Comparative studies of Lithosphere flexure on Earth and Venus

Diploma thesis of Ralph Mettier

Tutoring: Dr. K. Regenauer-Lieb, Prof. D. Giardini

Abstract

Our understanding of the Earth can be improved by the possibility to compare it to other

terrestrial planets. An important aspect of understanding the development of Earth-like

planets is the behaviour of the upper few hundred kilometres, which largely determines the

appearance of the surface of our planet. Of all planets in our solar system, only two remotely

resemble the Earth, namely Mars and Venus. While the surface conditions of Mars may be

closer to the conditions on the Earth, Venus is much closer to our home planet in terms of

size, mass and probably also composition. Therefore, Venus seems the prime candidate for

comparisons concerning the thermal aspect of the dynamics of the upper layers. In this

project, we compare a known region of lithosphere flexure on Earth with a possibly similar

region on Venus. For this a new method of wavelet analysis is developed and applied to

topography data. As a second step, the flexure example on Venus is modelled by a finite

element method. The applied model is mainly constructed around the Williams-Landell-Ferry

equation, a method that is adopted from polymer physics. It was shown that a model based on

this new approach, combined with the idea of au underlying plume is capable of producing a

fair fit over the inner rim, trench and outer rise of Artemis Corona. The resulting model

provides a promising starting point for further modelling of similar structures.

1. Venus

The second planet from the sun, and one of

the five original planets (originally

meaning wanderers) known since the

beginnings of history, Venus has often

been called the Earth’s sister planet. To

earthbound viewers, Venus is the brightest

object in the sky after the sun and our

moon, and has therefore always been

associated with beauty, and also with

women. Our current name for the planet

stems from the Roman name for the

Goddess of beauty and love, which they

adopted from the Greeks, where she was

known as Aphrodite. Because of Venus’

proximity to the sun, it appears to us

always within an angle of roughly 46.3 ° or

less of the sun. This value is obtained by

the following simple calculation:

1

E

VR

R=asin ( 1 )

with a the apparent angle between the sun

and Venus form our point of view, and RV

and RE as the orbital radii of Venus and

Earth respectively. This leaves three

possible states of observation:

1. Venus follows the sun closely,

thereby being best visible just

after sunset, when glare of the

sun is hidden by the horizon,

but Venus is still visible. This is

when Venus is seen as the

“evening Star”

2. Venus precedes the sun across

the sky, and is therefore best

seen shortly before sunrise,

where again, the glaring sun is

still shielded by the horizon.

This is the configuration when

Venus is seen as the “morning

Star”.

3. Venus is in front or behind the

sun form our point of view, or

to close to the apparent disc of

the sun to be seen against the

glare. In this state, Venus is not

visible to observers on Earth.

Venus therefore is always seen as morning

or evening star, which probably increased

its mythological appeal to early observers.

The same mechanism applies to Mercury,

albeit with a much smaller angle of

roughly 22.8°. Together with mercury’s

smaller radius, larger distance from Earth,

and much lower albedo, the effect of

mercury is much less spectacular.

Therefore, mercury is usually not even

noticed by most naked eye observers,

leaving the Morning/Evening Star title to

Venus.

Galileo Galilee is noted as having been the

first to train a telescope on Venus, and

thereby to notice that the planet goes

through phases, just like the moon. This

observation, along with his discovery of

the four largest moons of Jupiter, is to have

led him to embrace the Copernican system

of the planets revolving around a central

sun, in stead of the Ptolemaic model of a

stationary Earth circled by the rest of

creation, as preferred by the church at that

time. Later observers equipped with ever

better telescopes soon began to notice the

seemingly featureless surface of Venus.

What they were actually seeing of course,

is the dense unbroken cloud cover over the

planet, which is also responsible for the

planets extremely high albedo of 0.75

(NASA, planetary fact sheet – Venus), and

therefore for it’s extreme brightness in our

skies. By means of observing the planet

during transition in front of the sun, the

suspicion that the apparent surface being

just a cloud cover was confirmed, and the

2

first roughly accurate measurements of the

planets size were performed.

Unfortunately, Venus has no satellites,

meaning that the planets mass could not be

calculated exactly, and also it’s moment of

inertia was unknown, leaving basically no

method to obtain information about the

planets interior structure. After World War

II, the development of RAdio Detecting

And Ranging, RADAR, produced a

method for peeking below the clouds of the

shrouded planet (Goldstein et. al., 1965).

As first large surface structures were

identified, Venus surprisingly slow, and

retrograde rotation was noted (Carpente,

1970; Ingersoll et. al. 1978). Venus is the

only planet to rotate in a retrograde sense,

or, if following the definitions of

planetology, to have an axial inclination of

nearly 180 ° (The north pole of a planet is

defined as the pole above which an

observer must be positioned in order to

observe the planet rotating in a counter-

clockwise direction below him. The axial

inclination is then the deviation of the

rotational axis from a normal to the orbital

plane. Hence by definition, all planets

rotate in a counter clockwise sense, but the

retrograde orbiting Venus is ”stood on its

head”. Because of the absence of fixed

frames of reference in space, the two

definitions are equivalent.). The retrograde

rotation of Venus takes 243.7 Earth days to

complete, compared to it’s orbital period of

224.7 Earth days, making Venus “day”

longer than it’s year. The axis of Venus is

inclined by 177.36° to its orbital plane,

which for comparison is only 11.3% of

Earth inclination. Because of the planet’s

slow rotation, centrifugally caused

flattening is much less pronounced on

Venus than on Earth, making Venus an

almost perfect sphere. The rise of space

exploration brought a multitude of probes

to Venus, starting with Mariner 2 in 1962.

Venus was thus the first planet to be

reached by a space probe. Mariner

performed a 34’400 km flyby, collecting

data on Venus’ atmosphere and

temperature and also confirming the earlier

measurements of the planets strange

rotational period. Later, the Soviet probe

Venera 7 was the first to land on the

surface (Avduevsk et. al., 1971), and

transmit data. Due to the high

temperatures, and immense pressure at the

surface, the probe failed after transmitting

for only an hour. In 1982, Venera 13 was

the first mission to transmit colour

photographs of the surface back to Earth.

The NASA Pioneer Venus Mission was

launched in 1979, and entered orbit around

Venus, radar imaging the surface

(Pettengill et al., 1979; Dyer et al. 1974).

The latest mission to Venus was NASA’s

Magellan, described in detail in chapter 2.

The Magellan probe is also the origin of all

Venus topography data used for this study.

3

A global topographic map of Venus is

shown in figure 1.

As is known today, Venus has a mass of

4.8685 * 1024 kg, or roughly 0.815 Earth

masses. This mass is distributed over a

volume of 9.2843 *10 11 km3, giving the

planet a mean density of 5243 kg/m3, or

roughly 95% of Earths mean density. Since

the first interactions of artificial objects

and Venus gravity have been observed, it

was also possible to determine the moment

of inertia, which shows up at 0.33 which is

very similar to Earths value of 0.3308. This

implies an interior structure of similar

density distribution. Due to Venus smaller

size and mass, its gravitational attraction at

its surface is lower than that at Earths

surface, at 8.87ms-2. At Venus’ orbit,

which is closer to the sun, it receives a lot

more solar irradiation than Earth (2614

W/m2 compared to 1367 W/m2 for Earth),

but due to its high albedo (~0.75) the

surface planet actually absorbs less energy

from the suns radiation than Earth.

Nonetheless, the surface temperature of

Venus is extremely high, around 730 K,

which is higher than the average

temperature on mercury, the closest planet

to the sun. This high surface temperature is

reached by the mechanism of an extremely

efficient greenhouse effect, often deemed a

“runaway greenhouse”. The same effect is

well known on Earth, albeit to a much less

severe extent. This greenhouse effect is

caused by the planets very dense

atmosphere that is composed almost

exclusively of carbon dioxide (96.5% CO2,

3.5% N2, plus minor amounts of SO2, Ar,

H2O, CO and noble gases), which is known

to be a very potent greenhouse gas. The

pressure at surface level is around 90 times

the pressure at Earth sea level. Obviously,

under these circumstances, water is not

found on the surface, at least not in fluid

form. There are however water vapour

traces in the atmosphere. The slow

rotation, and dense atmosphere cause

winds on Venus to be very slow, in the

region of 0.6-1 ms-1. Although such slow

winds can move grains of surface material

from a few microns up to roughly a

millimeter in diameter, thanks to the high

density of the atmosphere, eolian erosion is

a small factor on Venus.

Geographical orientation on Venus is fixed

to radar bright feature named “Eve”. Eve is

a roughly ovoid patch of terrain that

because of it’s roughness reflects radar

signals from almost any incoming angle,

and therefore shows up as a bright spot in

radar images (figure 2). Eve is situated

slightly to the south of Alpha Regio, which

was the first surface feature discovered by

Earth bound radar. The bright patch Eve

caught the eye of the researcher, who used

it as the defining feature for the planets

prime meridian. Obviously, with the prime

median fixed, and the equator determined

4

by noting a planets axis of rotation, a

complete geographical grid of latitude and

longitude can be established. Due to the

fact that Venus has a retrograde rotation,

longitude increases to the east, unlike on

Earth. This is coherent with the definitions

of geographical grids layed down by the

IAU (Uchupi & Emery, 1993). Venus seen

globally, has two major highland

“continents”, surrounded by vast reches of

rather flat rolling plains. The main

highland areas are for one Ishtar Terra,

located near the northern pole area, and for

another Aphrodite Terra, located

equatorially at between 90 and 260 degrees

of longitude. Ishtar Terra is also the

location of the Maxwell Montes, the

highest region of Venus, with the highest

region reaching roughly 11km above the

datum, which is set at the planets mean

radius of 6051km. the Maxwell Montes are

named in honour of James Clerk Maxwell,

who hence became the only man to have a

feature on Venus named after him (Burba,

1990). All other features on Venus are

named after women, with the exceptions of

the neutrally named Alpha and Beta Regio.

These two large features are obviously

enough named for being the first and

second feature identified on the surface.

The lowest point on Venus is found in the

Diana Chasma, a deep trench like feature

in the Aphrodite area. Diana Chasma is

roughly 2km below the datum at its

deepest point. This gives Venus an overall

vertical relief of roughly 13 km, compared

to the Earths 20 km stemming from the

8850 m elevation of Mt. Everest (National

Geographic Society) to the –10’924 m of

the Mariana Trench. This smaller absolute

vertical range is typical of Venus, which is,

seen globally, a rather smooth planet. The

elevation histogram of Venus is compared

to Earths in figure 3. A further difference

in the two histograms is Venus unimodal

distribution of elevations (Ivanov et al.,

1996), compared to Earths bimodal

distribution reflecting continents and

oceanic basins. A similar histogram of the

elevations on Mars shows a trimodal

distribution, reflecting the northern

lowlands, the southern cratered highlands

and the Tharsis bulge and volcanoes

(Cattermole 1994). This unimodal

distribution in evidence of the vastly

different geological activity on Venus.

Assuming that Venus and Earth, due to

their similar size, density and moment of

inertia, have a similar internal structure,

and hence were probably very similar at

their accretion from the protosolar nebula

(Prinn et al., 1973), the question begs

itself, what made them so different today.

Both planets most probably had a similar

internal amount of heat to start with, but

while Earth developed plate tectonics as

the major mechanism of transporting heat

to the planets surface, and relies only

5

marginally on Hot Spots and mantle

plumes, Venus seems to do almost all it’s

cooling via these volcanic mechanisms,

and only very few structures are know that

might show some form of locally limited

subduction. One such feature is the main

objective of this project.

2. Magellan Mission

The topographical data used in this project

were provided by the North American

Aeronautical and Space Association,

NASA. They were recorded by NASA’s

Magellan mission. The following chapter

shall present some information on the

Magellan mission and the circumstances

under which these data were collected.

The Magellan Spacecraft which is shown

in figure 4 was launched on May 4th 1989.

It was the first planetary spacecraft to be

launched from the NASA space shuttle,

and not from it’s own surface to orbit

booster. After being released into low

Earth orbit by the space shuttle Atlantis,

the Magellan’s own inertial upper stage, a

solid fuel rocket propulsion unit, fired in

order to put the probe on the correct

trajectory to reach Venus orbit after a 15

month cruise. In order to achieve orbit

around Venus with a minimum burn and

thus maximum payload flight path, a

complicated path was chosen, swinging

Magellan one and a half times around the

sun. Magellan reached Venus on August

10th 1990 and inserted itself into a highly

elliptical polar orbit with a

periaphroditeum of 294 km and an

apoaphroditeum of 8’543 km. This orbit

was carefully calculated to have a period of

3 hours and 15 minutes, during this time,

Venus would have rotated under the flight

path by exactly the angle that the radar-

mapping aperture could image during the

periaphroditeum flyby. This is equivalent

to a strip of terrain 17 to 28 km wide. After

one Venus day, or 243 Earth days,

Magellan had mapped 84% of the surface

of Venus to an until then unknown

resolution. Until the end of the mission the

mapped surface had increased to 98% of

the planet. After the radar mapping was

done, the craft collected data on Venus’

gravity field by emitting a constant radio

signal directed at Earth. From the Doppler

shift of the signal, and from the

precalculated orbital ephemeredes,

deviations from the expected orbit could be

calculated, and a detailed gravity map

could be constructed (McNamee, 1993).

After May 1993, Magellan became the first

spacecraft to use the then new technique of

“aerobraking” (Curtis, 1994). By allowing

Magellan to graze the planets atmosphere,

the probe was slowed, lowering and

circularising it’s orbit (Lyons et al., 1995).

From the above-described highly elliptical

orbit, Magellan was manoeuvred into

6

amuch more circular orbit with 180 km

periaphroditeum and 541 km

apoaphroditeum. This gave the craft the

chance to collect much more detailed

gravity data on the high latitude regions of

Venus. The final experiment carried out

was purposely conducted at the price of the

spacecrafts life. Lowering the orbit further,

and placing the solar panels of the craft

into a so called “windmill” configuration,

the flight controllers were able to measure

the amount of torque developed by the

atmospheric friction, and thus to gain

information on conditions in the topmost

layers of the planets atmosphere. On

October 11th 1994, contact with Magellan

was lost in the Venus atmosphere, making

it the first orbiting probe to have been

crashed on purpose onto another planet.

All in all the Magellan mission was an

overwhelming success, yielding a vast

amount of data, that is to date still

unsurpassed in it’s quality and resolution.

With the Magellan mission, Venus became

the first planet besides the Earth to be

surveyed in such detail. All Magellan data

is freely available online for download at

NASA’s Magellan site.

3. Artemis corona

One of the most unusual geological

features that the Magellan data show on

Venus are the so-called coronae (Squyres

et al., 1992). These coronae are roughly

circular structures with diameters of 60 up

to well above of 2000 km with an average

diameter of 250km (Cattermole, 1994). A

few very large and prominent Coronae

were already identified by Venera 15 and

16 (Barsukov et al., 1984; Kotelnikov et al.

1984), as well as by earthbound radar

observations of the Venusian surface. With

the massive amount of altimetric data that

the NASA Mission "Magellan" provided,

several hundred more Coronae were

identified, the smallest with diameters

around 60 km. Coronae are very

widespread on the Venusian surface, 360

are known (Marov & Grinspoon, 1998),

covering roughly 49’000km2, and strangely

enough, they seem to occur only there.

They are as far as is known to date, unique

to Venus. Unlike craters, that also almost

exclusively of circular appearance, corona

are not impact structures.

The largest and most prominent of all

Coronae is Artemis Corona, a huge near-

circular structure with a diameter of

approximately 2600 km. The Corona

shows a vertical relief of roughly six km

from the deepest point of its surrounding

chasma, to the highest point of its inner

rim. The near circularity of the structure is

broken in the northwestern quadrant,

where mountainous topography either

overlays the corona, or prevented it's

forming. On a large scale, the corona

7

appears as a large flat disc, sagging

downwards in its middle, with a gradual

rise to it's highest regions around the rim.

Moving outward from the rim, the

topography drops very steeply into the

chasma, which surrounds the corona. The

chasma shows a maximum width of around

100km. Outwards from the chasma, the

topography shows a similar rise as along

the rim, albeit less highly elevated. This

outer rise is very steep on the inside, facing

the chasma, and gently curves back to the

elevation of the surrounding plains on the

outward facing side. The structure of

"ridge-trench-outer rise" optically reminds

strongly of subduction zones on Earth. The

fact that there are no coronae on Earth, and

so many on Venus suggests that these

structures are an important clue to the

different workings of planetary heat flow

of Venus and Earth.

The origin of Artemis and the other

coronae structures is still strongly debated.

Some popular theories describe the

coronae as mantle diapirs, or marks of

mantle upwellings. According to other

sources, the coronae could be retreating

circular subduction zones (Schubert and

Sandwell 1992). This is also adopted as a

working hypothesis for this project. The

development of a typical corona would

start with a relatively thin dense crust with

primarily elastic properties overlaying a

less dense viscoelastic mantle. The density

contrast can stem from thermal differences,

as well as from phase changes in the

material. Through some mechanism

(plume, simple failure, meteorite impact,

volcanic activity, etc.), the thin dense crust

ruptures at one point, causing the crustal

material around the fault to sink into the

soft and hot mantle material below. The

downward pull from the submerged

material is then sufficient to drag further

material downward, starting a circular

subduction zone, which constantly retreats

with a radius of curvature decreasing with

time. Mantle material is pushed up in the

centre of the circular subduction zone,

cooling to form a covering "disc". As the

structure expands, the main region of

newly surfaced mantle material is relayed

to the sides, where the subduction is still

continuing, thereby reducing the support of

the older cooler material in the centre of

the disc. Because of this failing support,

and the increased density due to the

cooling, the centre of the disc starts to sag

downwards. This procedure can continue

until the hoop and bending stresses along

the subducting edge of the crustal material

balance the downward pull of the

submerged material which in turn is

controlled by thermal diffusion. This must

happen at some point, as the amount of

subducted material compared to the

amount of material still at the surface

decreases constantly due to the geometry

8

of the feature. Therefore it stands to

reason, that a corona has a maximum size

it can reach, and that this maximum size is

dependent of the properties the upper

mantle and plate in the region of the

corona.

4. Earth topographic data

All the topographical data describing land

surfaces of Earth used in this study were

taken from the GTOPO30 dataset. This

dataset was augmented with the

bathymetry dataset "topo_8.1" provided by

David T. Sandwell. Together, these

datasets give us a two minute of arc

resolution topographic map of the Earth’s

surface between 72 degrees latitude, north

and south.

The bathymetry data originates from a

recent publication (Smith and Sandwell,

1997) and is derived from satellite

altimetry measurements combined with

selected shipboard echo soundings.

The GTOPO30 dataset is a global digital

elevation model with a horizontal

resolution of 30 seconds of arc (hence the

name). The data was compiled, from

several raster and vector sources of

topographical information, in 1996. The

dataset’s resolution was scaled down to

two minutes of arc in order to correspond

with the bathymetry data mentioned above.

A resolution of two minutes of arc

corresponds to roughly 3.7 km at Earth’s

equator, which should be more than

detailed enough for our purposes,

considering that we are working with

structures of several hundred km

wavelength.

The combined, bathymetry/GTOPO30

dataset was used as distributed by

Sandwell, and shall in this project hereafter

be called "Sandwell data".

The Sandwell data comes packaged in a

file named topo_8_2.img, which serves as

the input for a Matlab routine

"mygrid_sand.m" which is also available

from the Smith and Sandwell website. This

Matlab script provides rectangular sections

of the global dataset when supplied with

the north, south, east and west boundaries

in degrees. Unfortunately, the script is

unable to provide a global "readout", so

that the data had to be extracted in

packages by a additional script of our own

working. These packages were then pasted

together to create one rather large data

matrix with values representing the

elevation at each point of the Earth

between 72 degrees north and south (see

figure 5). Projection is given by the

original Matlab routine to be Mercator. A

perspective correction of the data's aspect

ratio on hand of the projection was not

considered necessary.

9

5. Mexican Hat wavelet analysis

In order to compare extents of lithosphere

flexure of different structures, it is

important to be able to identify the regions

where the topography reflects the

lithosphere flexure and to determine the

wavelength of these structures. Together

with the amplitude of the flexure, this

presents a method of quantifying one

example of flexure in order to compare it

with an other example. Identifying these

regions is fairly straightforward. Simply

looking at a topographical or bathymetrical

map will let you notice several areas,

where there is a noticeable rise in

topography that follows a trench or other

structure known to cause flexure. As an

example, the topographic image of the

Aleutian trench can be seen in figure 6.

Similarly, a significant outer rise is visible

on the topographic image of the Artemis

region in figure 7. But as easy as it is to

identify these rises, it’s quite difficult to

compare them. Their wavelengths are

rather difficult to determine just by eye

judgement, mainly because they blend into

the surrounding plains asymptotically , and

because they are overlaid by other

structures, such as mountain ranges. In

some cases, the rises can be measured

more easily in a profile of the structure, as

is seen in figures 8. An elegant method of

identifying these structures is wavelet

analysis. The method of wavelet analysis

has been used before on two dimensional

datasets (Malamud and Turcotte, 2001),

and is basically rather simple. If a data

function, for example a topographic

profile, is convolved with a suitably shaped

wavelet function, a new function is

obtained. This function shows the

similarity of the two initial functions at

every point of the grid. If the chosen

wavelet function is shaped like a typical

topographical feature, the resulting

function will have high values in the region

where such a structure is present in the

topographical data. By performing the

convolutions with several wavelets of

similar shape, but different wavelengths,

the wavelengths of an existing structure of

this type can be determined, being the

wavelength of the wavelet that produced

the highest values in the convolution. This

technique has been applied for example to

determine overall roughness of terrain

profiles, or dominant wavelengths of

topography on Mars (Malamud and

Turcotte, 2001). What we introduce here is

a similar technique, albeit used directly on

the 3D topography datasets of Earth and

Venus used for this study. The method is

basically the same, only the chosen

wavelet has to be changed from a two

dimensional function to a three

dimensional analogue. This is easily done

10

by rotating the wavelet around it’s

symmetry axis. Figures 9 and 10 show the

change from two to three dimensional

wavelet. This wavelet matrix is then

convolved with the topography data in

order to obtain a “similarity matrix”

showing the degree of fit between the real

topography and the synthetic wavelet. This

is repeated for several wavelets of different

wavelengths, and the wavelength of the

wavelet that produced the best fit is then

considered the dominant wavelength at

each point of the topography. In practice,

some technicalities must be considered:

- A convolution between a m*n

matrix (topography) and a k*k matrix

(wavelet) produces a (m+k-1)*(n+k-1)

matrix. The representation of the original

topography matrix is surrounded by a k/2

element wide band of values that are

generated by the convolution of the

wavelet matrix with the edge of the

topography matrix. This band must be

removed from the resulting matrix before

the maximum is determined. This

phenomenon is known as the boundary

effect, and is described in more detail at

the end of chapter 7.

- The wavelet matrix is not a

continuous representation of the wavelet

function, but a discrete series of values.

Choosing the wavelength of the wavelet

matrix too small can result in the wavelet

“losing shape”. Taken to extremes, the

supposed wavelet is reduced to a singular

spike in the centre of a null matrix.

Convolution with these disfigured wavelets

often returns incorrectly high values.

Therefore, attention must be paid to using

data with a high enough resolution for the

method to work.

- The CPU time and memory

needed for convolving and manipulating

matrices of such large dimensions is not

practicable for a process that must be

repeated so many times. Therefore, it is

advisable to reduce the size of the input

matrix (topography) by using a square

averaging method. This reduces the

resolution of the topography data. Also, the

resulting wavelengths must be multiplied

with the smoothing factor used in order to

obtain the correct wavelengths.

- In regions where the flexural

structures are strongly covered by short

wavelength topography, the best fit will

inevitably result from shorter wavelength

wavelets, masking the structure that was

originally of interest. This is to some

degree related to the problem of short

wavelengths noted above.

- Large wavelength wavelets

produce large boundary effects from their

convolution with the edges of the

topography matrix. If the input matrix is

chosen to small, these boundary effects can

dominate the whole resulting matrix. This

problem can be avoided by choosing the

11

input matrix significantly larger than the

dimensions of the structure to be identified.

As the boundary effect is created by

convolving the wavelet with the last

sample of the data matrix, the created

disturbance has half the width of the hat

matrix. therefore, extending the data matrix

by slightly more than half a hat size in all

directions is necessary.

6. Why the Mexican Hat ?

Choosing a suitably shaped wavelet for the

method described above is of supreme

importance. The shape should be similar to

the expected topographical structures,

should be radially symmetric, must

integrate to zero and must be

mathematically easily handled in order to

produce several wavelet of the same type

with different wavelengths. Our choice fell

on the Mexican Hat Wavelet. The Mexican

Hat is mathematically the negative second

derivative of a gaussian probability density

function (bell curve). If can be described

by the following formula:

( ) ( ) 2241 2

13

2 xexx -- -öö

÷

õææç

å=Y p ( 2 )

The Mexican Hat therefore belongs to the

large family of gaussian derivative

wavelets, in general grouped under the

name “vanishing momenta wavelets”. It is

well suited for our purposed because it is

symmetrical, and can therefore easily be

rotated in order to create the needed three

dimensional wavelet matrix. Also,

geological structures are often shaped by

erosional processes. Erosion is

mathematically described as the

convolution with a gauss curve, so a

gaussian based wavelet will show more

similarity to a geological feature than a

non-gaussian wavelet. Finally, a further

reason for our choice in favour of the

Mexican Hat Wavelet was the fact that the

function is already implemented in the

Matlab wavelet analysis toolbox, and was

therefore easily available and did not

require the writing of additional scripts.

A difficulty in using the wavelet is that the

standard form of the Mexican Hat

represents less than one full oscillation,

and therefore, the definition of its

wavelength is rather arbitrary. This is

however not really a problem because we

are not looking for absolute wavelengths,

but for a method to compare wavelengths

between examples of the same

topographical structure. We therefore

chose to define the wavelength of the

Mexican Hat as the length between the two

zero crossings as shown in figure 9.

Actually, the function has two other zero

crossings, at plus and minus infinity, as it

approaches the x-axis asymptotically from

below. The crossings mentioned above are

12

obviously meant to be the first two, at, or

near to, one and minus one. Translated to

the analysis of the topographical features,

what we describe as the wavelength of a

feature is actually closer to the half

wavelength. Because many topographical

structures only show a concave half period

or a convex half period, and rarely a

complete cycle, this is a useful length to

work with. The relation between true

wavelength of a flexure and our definition

of wavelength is shown in figure 11.

7. Mexican Hat Analysis of the Aleutian

Trench

The search for an analogue structure to the

Venusian coronae in general, and to

Artemis Corona in particular is rather

difficult. Generally, coronae do not exist

on Earth, in fact they seem to be uniquely a

Venusian phenomenon. A possible

candidate, at least for similar lithosphere

flexure and trench dimensions are the

Pacific subduction zones. The Aleutian

trench subduction zone was chosen as a

probable candidate for well identifiable

lithosphere flexure. The Aleutians are a

typical back arc island chain, reaching

from the south western tip of Alaska across

the north Pacific Ocean towards the

Russian peninsula of Kamtschatka. The

whole structure of trench, island chain and

back arc basin are part of the often

mentioned “ring of fire” around the Pacific

rim. Politically the islands are part of the

US state Alaska. The islands are all

products of the typical back arc volcanism

that occurs behind a subduction zone as is

shown schematically in figure 12.

Seawards (south) of the island chain, the

Aleutian trench is formed by the northern

rim of the Pacific plate being subducted

under the North American plate. The

trench formed by this subduction process is

several kilometres deep. Further seawards,

the lithosphere is lifted upwards several

hundred meters forming a typical outer rise

that parallels the trench. This system of

trench-outer rise is easily identifiable

simply by studying the topographic image

of the region shown in figure 6. It becomes

even more visible in a plot which shows

terrain tilt in N/S direction, the map in

figure 13 was created by NOAA and is

available at:

http://www.ngs.noaa.gov/GEOID/IMAGE

96

Because of the strong east-west

orientation of the trench, and therefore also

of the outer rise, the tilt is much better

visible in this map than in a east-west tilt

plot. Even though the structure is easily

seen in these maps, the determination of its

wavelength is difficult, mainly because it is

difficult to determine where the rise ends,

and the normal elevation of the seafloor

13

begins. This is the reason that the data is

subjected to a Mexican Hat Wavelet

analysis.

The first step was to read the data from the

“topo_8_2.img” data file. More details on

the data, origin and file format are given in

the chapter “The Sandwell data”. The data

of the whole file was extracted and

combined into a complete map of the

Earths topography between 72 degrees

northern and southern latitude. For this

task, a Matlab script (“makeEarth.m”) was

written that divided the task into 5 degrees

of longitude wide swaths that were

extracted by the “mygrid_sand” script and

then joined to form a world map. From this

map, the region of interest was selected,

and cropped from the world map. This

segment of topographic data still had the

original resolution of the dataset, which

converts to around 3.61 km per pixel. This

is obviously an uncomfortable resolution

for further work, so the data was regridded

to a 1km per pixel matrix by the Matlab

routine “interp2”. The “griddata” routine,

which was originally designed for this task,

was unfortunately not usable, because of

insufficient memory resources. Even

though the amount of data is dramatically

increased by this regridding, it is important

to remember than the information content

is not increased. This km grid dataset is not

directly used for the analysis, but just as a

basis archive, from which any other

resolution can be constructed. The analysis

is performed by a further Matlab script

(“mexearth.m”). This script first uses a

smoothing procedure to decrease the size

of the data tensor (“ImageFlat.m”).

Because we are looking for structures of

many tens to a few hundreds of km in

dimension, reducing the data to 10 or 20

km resolution does not hamper the analysis

as can be seen in the comparison of the two

maps in figure 14. Actually, the smoothing

effect also acts as a low pass filter,

removing short wavelength topography

quite effectively, and thus increasing the

probability of finding the structures we are

looking for. It is also important for the

analysis to not exaggerate vertically!

Exaggerating the data vertically changes

the overall shape of the topography in

respect to the wavelets. As a second step,

the “Hats” are generated. For a given list of

wavelengths, 3 dimensional Mexican Hat

Wavelets are calculated, to matrix size that

was chosen to work well with the size of

the data matrix. In the case of the Aleutian

trench dataset, the Hats were chosen as a

linearly spaced series with between 20 and

150, with 20 increments. After the

analysis, the lengths had to be remultiplied

with the smoothing factor of 4, which was

chosen in order to reduce the amount of

data.

In the resulting map, shown in figure 15,

the trench and the outer rise are easily

14

visible as continuous band of dominant

wavelength. The continuous orange/red

stripes portrait the island chains, which of

course all have similar sizes and

amplitudes, and therefore also reflect in

their typical wavelength. The dark blue

stripe shows the actual Aleutian trench,

which is surely the most dominant element

in this plot. In the case of the outer rise,

shown as the slightly broken yellow band,

this is the wavelength of the upward

lithosphere flexure that we were looking

for. The wavelength is shown to be in the

region of 360km which agrees well with

what can be judged from the profile shown

in figure 11. The analysis of topography

data with the Mexican Hat Wavelet

method developed here is a promising tool

for further analyses of flexural rigidity.

Problems that showed up during the use of

this procedure are the following:

- Terrain with high roughness

in short wavelength can apparently mask

the long wavelength structures that are the

key to this project. This is to be expected,

as the method was designed to determine

the dominant wavelengths in a region, and

not to filter out structure of a special

wavelength. Choosing the wavelength

range of the wavelet “hats” to exclude

these short wavelengths is a possible

method of minimizing this effect.

- Using long wavelength

“hats” in order to identify structure if long

wavelength topographical structures causes

larger boundary effects, where the wavelet

overlaps the edge of the topographical

data, and must therefore be compensated

by choosing a much larger section of

topography data as input matrix for the

analysis script. This means basically a

much larger amount of data to be

processed, and therefore drastically

increased computational time. This effect

can be slightly compensated by calculating

fewer hats, but this requires a better fore

sight of which wavelengths are required.

Also, a hypothetical combination of rather

short and very long wavelengths would

require very extensive calculations.

8. Mexican Hat Analysis of Artemis

Corona

The wavelet analysis performed on

Artemis Corona was basically the same as

the procedure described in the previous

chapter, dealing with the analysis of the

Aleutian trench. The same script was used

and slightly modified (“mexvenus.m”).

The data was taken from the Magellan

topography data described in chapters 2

and 3, and the Artemis region was cropped

out of the 1km resolution data as a

rectangular section 4491km by 6050km in

size. Due to the much larger data volume,

the smoothing was performed over squares

of 10km. This was again done with the

15

“Imageflat.m” script, and rendered a

dataset with a more suitable extent. The

stronger smoothing due to the larger factor

chosen should not be a problem, because

we are expecting larger structures anyway.

The “hats” were chosen in a series of 20

linearly spaced values between 10 and 120.

Compensated for the smoothing factor of

10, this yields lengths between 100 and

1200 km. Considering the 1300km

diameter of the Artemis structure (see

profile, figure 8), and the few hundred km

width of the chasma, these lengths can be

considered appropriately chosen.

The resulting map shown in figure 17

surprisingly gives a less clear picture than

the analysis done for the Aleutian trench.

Nonetheless, the chasma and out rise are

quite easily determined as the pale/dark

blue band (chasma) and the yellow/green

band (outer rise). The wavelengths of the

chasma shows up at the expected length of

350-450 km, whereas the outer rise seems

to be dominated by 700 to 800 km lengths.

this enables us to directly compare the

wavelengths of the two flexures. As

expected, the flexure around Artemis is

larger by almost half an order of

magnitude. This corresponds well to the

unnaturally high downward pull forces that

had to be applied to the model to achieve

similar structures (see chapter 19).

9. Modelling Artemis Corona

As was shown in the previous chapters, the

lithosphere flexure in the region of Artemis

Corona displays a strong similarity to the

flexures observed along subduction zones

on Earth. The main difference is in the

scale of the structures, and in amplitude.

The point of the following chapters is to

document our attempt at modelling an

Artemis like structure by simulating a

hypothetical circular subduction, and

changing the parameters in order to obtain

a model of similar dimensions and profile

as the real Artemis Corona.

Our model was designed to be a fully

viscoelastic model, using only one set of

material parameters. Determining two

different materials, with different

properties for plate and mantle would

probably have facilitated obtaining the

desired shapes, but would have been less

realistic. The determining parameter of our

model was supposed to be temperature.

Density of the material is directly coupled

to its temperature, as are its elastic and

viscous properties. The initial temperature

field was determined by the known surface

temperature for Venus, and an estimate for

a suitable temperature gradient (Sandwell

& Schubert, 1992).

In order to try different variations of the

starting parameters, a “standard” model

was developed, which gave qualitatively

16

the correct shape. This standard model is

described in chapter 14.

Basically, the model consists of a

rectangular block of material, with

temperature increasing as a linear function

of depth. The size of this block is chosen to

reflect the size of the actual feature. On the

surface is a superimposed cold layer of

material, which is subducting near the

centre of the model. An image of the

standard model in it’s starting conditions

can be seen in figure 18. The elastic

properties of the cold plate material should

cause the plate to flex upward, forming the

distinct outer rise seen around Artemis.

Directly above the subduction zone, the

surface should be drawn downwards into

the models version of Artemis Chasma.

Further left, the subducting material should

displace the underlying mantel material,

causing the surface to bulge upwards in

imitation of the Artemis rim. Plastic

properties and failure criteria of the

material were not used, which is probably

responsible for discrepancies between

model and real corona in the chasma.

10. The Williams-Landell-Ferry

Equation

The viscoelastic properties of a planetary

material are usually considered to be a

material constant. This is however not so

for large temperature changes, and also for

pressure changes. A similar viscoelastic

material is a polymer which experiences a

phase transition at the glass transition

temperature associated with a rapid drop of

elasticity over several orders of magnitude.

The behaviour of this viscoelastic material

at the so called glass transition temperature

can be described by a time shift in the

relaxation time given by:

( )( )02

01 *log

ttCttC

aT -+-

= ( 3 )

This is known as the Williams-Landell-

Ferry Equation (Ferry, 1980). In the above

equation, aT is the above mentioned shift

factor at the temperature t and t0 is the

reference temperature, usually the glass

transition temperature where the material

stops behaving elastically, and starts to

become dominantly viscous. The C1 and C2

values are two parameters that determine

the steepness and curvature of the function

shown in figure 19. For our purposes, the

higher value of C2 was set as a fixed value

of 1000, and the C1 value was changed as a

model parameter, as described in chapter

15. It should be noted that in the time

integrated F-E description rapid drops of

viscosity (the mantle) or elasticity (the

polymer) are formally equivalent in their

effect on the relaxation time.

11. The temperature field of our model

17

The temperature distribution of our model

is based mainly on the known average

surface temperature of Venus and the

lithosphere temperature gradient of 4.8 –

3.2 Kkm-1 given by Sandwell and Schubert

(1992). The gradient used in their work is

based on the calculations of elastic

thickness of the lithosphere from the

curvature radii of flexure zones. The

average surface temperature of Venus is in

the region of 728 K, with very small

variations during the year, due to the fact

that there are hardly any mentionable

winds at surface level on Venus. From the

surface temperature and the temperature

gradient, a smooth depth dependant

temperature field is generated, the

temperature of the plate and the subducted

plate material is then superimposed on this

smooth field. Basically, there is a feedback

mechanism at work, caused by the thermal

conduction of heat from the warmer

surrounding material to the cooler

subducted material. Because of the

geologically short timescales modelled in

our case, these are however hardly

noticeable in the model. Nonetheless, the

specific heat capacity and thermal

conductivity of mantle material was set at

1500 and 3, which agrees with values

obtained for mantle material of the Earth.

These are probably only approximations

for Venus, but without direct information

obtained by possible future landers, they

are our best guess.

12. Gravity change with depth

Due to the fact that an average planet does

not have homogenous density distribution

with depth, gravity will not decrease

constantly towards the centre of the planet

as it would in a hypothetical homogenous

sphere. The gravitational pull towards the

planets centre is only influenced by the

mass contained in the sphere with radius r,

with r being the current distance from the

centre of the planet. This can easily be

shown by a geometrical observation. First,

in a two dimensional shell with

infinitesimally small thickness, or simply

said, a circle, it is obvious that any possible

acceleration on a point inside the circle due

to the mass of the shell cancels out with the

attraction from the portion of the shell

diametrically opposite the attracting

portion. The inverse square law of gravity

exactly cancels out the increased mass due

to the increased area inside the angle of

view. As a sphere can be considered an

array of many circles, rotated around a

common diameter, the same must apply for

the inside of the sphere. A shell of finite

thickness can again be built up of an

infinite number of zero thickness shells, so

18

that by reasoning, the zero acceleration

rule must apply to the enclosed area of any

hollow sphere

The gravitational attraction of the planet at

a given distance from its centre can be

calculated with the following equation:

( ) ( ) ( )ñ-=-=r

dxxxrG

rrmGrg

0

222 4 rp ( 4 )

with r(x) as the density function of depth.

One of the most common depth/density

models used for Earth is the PREM model.

If the PREM model is inserted as the r(x)

density function into equation 4, the

gravitational attraction of the planet stays

remarkably close to its surface value of

9.81 ms-2 throughout the mantle. For the

lower 600 km of the mantle, the value

increases to its overall maximum value of

10.8 ms-2, which is reached at the core

mantle boundary. Below the core mantle

boundary the gravitational acceleration

towards the centre decreases constantly,

albeit at a higher rate than it would for a

homogenous Earth, rather like expected for

a homogenous sphere of higher density

than Earth average. Due to the fact that

gravity stays practically the same for the

top 2000 km of the mantle, a model with a

depth of 420 km depicting exclusively

mantle and crustal material can easily

ignore gravity changes with depth.

The degree of exactness when applying a

depth/density model like PREM to Venus,

even though the model was developed for

Earth stands to debate. Obviously Venus

cannot have identical rheological

properties as Earth, and therefore the

model will be a loose fit at best. But

considering that Venus is of similar size

and mass as the Earth, and that their

inertial moment of rotation is very similar

(Venus 0.33, Earth 0.3308), it can be

suggested that their internal structure is at

least similar. The higher surface

temperature and slightly smaller radius of

Venus suggest that the top 420 km of the

mantle might correspond roughly to a top

section of Earths mantle. Because of the

comparatively small vertical size of the

model, it can be assumed that the 420km of

the model are well with the region of

constant gravity.

13. Gravity Change with Latitude

On a rotating planet, the gravitational

attraction towards the centre is

counteracted partially by the centrifugal

force created by the planets rotation. This

centrifugal component is obviously at its

maximum at the planets equator and zero

at it’s poles. The size of this centrifugal

acceleration can be calculated from the

planets radius, and its period of rotation by

the following equations:

19

Actual radius r of the described circular

path:

jcos*Rr = ( 5 )

with R as the planets mean radius, and j as

the latitude of the point of interest. The

centrifugal acceleration at this point would

then be:

rac2w= ( 6 )

where w is the angular velocity. On Earth

this value could be neglected compared to

the gravitational acceleration. On Venus,

where the rotational period is very much

longer than on Earth, the centrifugal

component becomes very much smaller.

On the basis of this conclusion, and the

conclusions on gravity change with depth,

gravity can safely be accepted as constant

in our model of Artemis corona.

14. The „standard“ model

The axissymmetirc modelling of a block of

one material in order to achieve a similar

surface topography as seen in Artemis

Corona on Venus and described in chapter

3 requires the choosing and subsequent

fine-tuning of several material parameters

and geometrical quantities. Starting from a

“standard” model that is based partially on

parameters defined by earlier publications

about similar structures on Venus and

Earth, and partially by arbitrarily defined

values for other parameters, the model is

run with several modifications to the value

of a single parameter in order to choose the

best suited values. The order in which

these parameters were selected is largely

determined by trial and error. We have

purposely selected a permissible parameter

range beyond reason to test the robustness

and uniqueness of the best fit parameter.

Also, if the best fit is obtained by a

parameter or geometrical quantity that does

not make sense from what we know of the

Earth, we would conclude that the basic

assumptions are incorrect or the new WLF

method is inappropriate. The order in

which the parameters are adjusted certainly

has some influence on the values that are

finally obtained, but as we have no way of

determining which of several equivalent

combinations of parameters is actually the

most realistic, we will accept the outcome

of the chosen order to be the overall best

fit, to be improved in future studies. The

standard model that all model runs are

compared to is defined by the following

parameters (described in the order used as

input for the Matlab function

“makeinp.m”):

Model dimensions: - width

These parameters were

chosen mainly to be comparable to the

dimensions of the real object, Artemis

20

Corona on Venus, described in chapter 3.

The real corona is near circular, and of an

average diameter of roughly 2600km.

According to our favourite theory of the

coronas origin, it is or once was an

expanding circular subduction mechanism

(Sandwell & Schubert 1992). It can be

argued that the structure, which represents

the largest of its kind on Venus, has

reached the maximum diameter possible

for such a structure and has therefore

stopped expanding. The dynamically

expanding nature of the structure is

however a key factor to the understanding

of this mechanism. The length of the

model was therefore chosen to depict a

younger smaller Artemis Corona, with a

diameter of only 2400 km. The

topographical profile extracted from

Artemis is however taken to represent the

topography of this 2400km corona. The

difference in diameter between the original

and model corona amounts to about 8%. It

is probably safe to assume that this small

change in diameter should have no drastic

change on the rim-chasma-outer rise

topography. The 2400 km width of the

model was placed with the chasma at the

centre point, with the intent of having

enough space outwards of the rim to

correctly model the outer slope of the outer

rise. The model width is represented in the

Matlab scripts by the variable “width”.

-Depth

The depth of the model was chosen

at 420km. This value was chosen rather

“out of the blue” with consideration made

to the following properties.

The depth as also the width should

be whole number multiples of a maximal

number of possible block sizes. A value of

420 km is obviously dividable into many

more possible block sizes than a round

value of i.e. 400km (4, 5, 6, 7, 10, 12, 14,

15, 20, 21, 28, 30 … as compared to 4, 5,

8, 10, 16, 20, 25, 30 …).

We do not intend to model any

deep mantle properties or mantle core

interactions. Therefore, the depth should be

substantially smaller than the mantle depth

of Venus, which is roughly 2800km

(Marov & Grinspoon 1998). Additionally

temperature/depth models of Venus

(Marov &Grinspoon 1998) show a

discontinuity in smoothness of the

temperature curve at roughly 900km depth.

As our model disregards phase transitions,

it should avoid deeper mantle regions.

Larger models with the same block

size require massively more computational

time. As we only subduct material to a

maximum depth of 300 km, a 100 km deep

undisturbed zone seems sufficient to avoid

boundary effects from the model floor on

the subduction process.

Block size:

21

The size of the blocks used for

modelling is a key factor. Generally

smaller blocks are preferable, simply

because they give a higher resolution, and

more realistic material behaviour.

Unfortunately, choosing a smaller block

size is a very efficient way of increasing

computational time for the model.

Therefore, blocks are chosen as large as

possible without creating problems of

artefacts in the model. Due to our choice of

width and depth, a large number of block

sizes ranging for 4 to 30km very possible.

An important factor in choosing the

suitable block size is the thickness of the

surface layer of cold material that is used

for the model. This plate should definitely

not be thinner than one block. Therefore,

plate thickness as determined by earlier

research (Sandwell & Schubert 1992) can

be considered an important criterion in

choosing suitable block sizes. For the

standard model, a block size of 15 km is

the lowest quality compromise between

resolution and fast running model. This can

be changed during the modelling process

in order to achieve a better fit.

Surface temperature:

-plate

In order to maintain a density

contrast between the surface material

outside the corona and the material inside

the corona, the temperature of the material

had to be defined separately for the two

areas. The real average surface temperature

on Venus is 728 K (Schubert &Sandwell

1992, Cattermole 1994). This is the

temperature used for the cold surface of the

plate. This choice was made because the

region outside of the corona makes up a 3

times larger area in our model than the

inside of the corona, therefore, the error is

minimized somewhat. The temperature of

the plate is given in the Matlab scripts

under the variable “surftemp”.

- corona

The material inside the corona must be set

with a higher surface temperature than the

plate material in order to obtain the lower

density that is necessary for the subduction

process to work. In the standard model,

this value is set to 1000 K, which is very

much higher than the surface temperature

in the area because of the WLF

approximation. The value of 1000 was

chosen in order for the material to behave

like mantle material, in a mainly viscous

way. The reference temperature in the

WLF equations described in chapter 10 is

also chosen to be 1000 K. This is the

temperature that should be present at the

base of the lithosphere, according to the

smooth temperature gradient assumed

(Schubert & Sandwell 1992).

Plate thickness:

22

The thickness of the zone of cold

surface material is a major factor in the

elastic behaviour of the plate. A thick plate

will less easily bend under the pull of the

subducted material, thereby changing the

wavelength and amplitude of the outer rise.

A thicker cold, and therefore dense, plate

lying on top of the model also increases the

pressure to which the underlying warmer

material is subjected. The available

thicknesses that can be applied in the

model are strongly limited by the blocksize

that was chosen. Obviously the thickness

of the plate can only be a whole multiple of

the blocksize. This means that basically,

the decision is down using a one or a two

blocks thick layer of cold material. The

value used in the standard model is one

block thickness. Because of the short time

available for modelling (< 2 months) this

approach was chosen as a rough

approximation at best. Clearly, a more

refined analysis is necessary. Together

with the standard models block size of

15km, this gives a suitably thick plate that

is roughly in accordance to the lithosphere

thicknesses used in previous work

(Sandwell & Schubert 1992, Marov &

Grinspoon 1998, Solomon & Head 1991).

The plate thickness is represented in the

Matlab scripts with the variable

“platethick”.

Length of subducted slab:

The main “motor” of the

subduction process modelled is the

downward pull of gravity on the subducted

slab material as is shown schematically in

figure 12. As the initiation of the

subduction process is not part of this study,

the fact that subduction takes place is

accepted as given. The subduction process

is modelled as having begun long ago, and

to keep running to the time of the model.

For this effect, the subducted slab is

applied, like the plate, specifically onto the

smooth temperature field, at a given

starting temperature. In order to determine

which elements of the model are part of the

slab and must hence have their

temperatures changed, the “makeinp.m”

script needs two parameters: Subduction

point and length of the slab. The angle at

which the slab descends is fixed to 45°.

This is a value that was picked mainly

because of its usefulness in calculations.

However it is well within the margin of

subduction dips determined in other studies

(i.e. Creager & Boyd 1991). Because we

are only modelling the upper 150km of the

subducted plate, the rather low value of

45° is acceptable due to the fact that

subduction angles usually get steeper with

depth. As the subduction happens at the

centre of the model, the only required input

parameter is the length of the subducted

slab. This length is given in the variable

23

“sublength”. The standard model value for

subducted length is 150km.

Density of the subducted material:

In order to create a downward pull

that is strong enough to cause the

magnitude of lithosphere flexure we were

looking for, a significant mass must be

appointed to the subducted slab. We would

like to point out that our model is still in

the same group of “flexural rigidity

models” despite the fact that that the

mantle is modelled explicitly. Hence our

effective elastic plate thickness

underestimates the real plate thickness. In

order to generate the same body force as a

thicker slab, the density of the effective

elastic layer has to be increased artificially.

Because our model couples density directly

to temperature, the corresponding density

is freely adjustable by arbitrarily varying

the temperature of the slab. The standard

model uses a density of 8000 kgm-3. This

value is represented in the Matlab scripts

by the variable “subdens”. Together with

the above mentioned length of the

subducted slab, this parameter determines

the amount of mass “causing” the

downward pull. Adjusting these parameters

is a key feature in fitting the amplitude of

lithosphere flexure.

Plate density:

The density of the cold surface

material forming the plate is one of three

parameters determining the amount of

mass that is contained in the plate and

therefore the pressure to which the

underlying mantle material is subjected.

The value of 3330kgm-3 that is used in the

standard model is a rather high estimate

based on the values for Earths upper

mantle and lithosphere. The chosen value

is used in the “makeinp.m” script to

determine the temperature density

function. It is thereby coupled with the

plate temperature parameter allowing

material that is exposed to the surface to

reach this density after cooling. The value

is represented in the scripts by the variable

“platedens”. And is allowed to vary by

about 50kgm-3 depending on temperature

variances inside the plate.

Mantle density:

The counterpart of the plate density

in creating a density contrast is of course

the density of the underlying mantle

material. This density must be lower than

the density of the plate material in order to

achieve the negative buoyancy needed for

a subduction process. The value used for

the standard model is 3250 kgm-3. This

value is represented in the Matlab scripts

by the variable “mantledens”.

Poisson ratio of: -plate

24

The Poisson ratio is a temperature

dependent material constant that describes

a key elastic property of a material. The

Poisson ration is defined as the ratio of the

transverse contraction strain to longitudinal

extension strain in the direction of the

stretching force. Thereby, tensile

deformation is considered positive and

compressive deformation is considered

negative. The Poisson ratio is defined to

contain an additional minus sign in order to

assure that normal materials have positive

Poisson ratios. Virtually all natural

materials have a positive Poisson ratio,

which mean they become thinner when

stretched. The standard model value for the

Poisson ratio of the plate is set to 0.25,

defining an elastic material that is equally

compressible and deformable. Earlier

studies (Sandwell & Schubert 1992) used a

Poisson ratio of 0.25 for the whole of their

models, when performing purely elastic

modelling of Artemis Corona and other

features on Venus. The value used in our

models is taken directly from this source.

-mantle

The Poisson ratio of the mantle is

distinctly different from the ratio of the

plate material. The mantle material

underlying the cold surface plate should

behave in a more liquid way than the plate.

A completely incompressible liquid would

have a Poisson ratio of 0.5, meaning no

loss of volume under pressure, and purely

deformation. Such a material is

compressible to zero height, redistributing

all its volume to the sides, as is typically

observed in liquids. Mantle material

however is not an ideal fluid, and therefore

shows compressibility, and also a certain

resistance to deformation. In order to

achieve suitably viscoelastic behaviour of

the mantle material, a Poisson ratio of 0.35

was chosen for the standard model value,

and is represented in the scripts by the

variable “mantlepois”.

Young’s Modulus:

Young’s modulus is another

material constant that describes the

necessary normal stress needed for a

proportional deformation. Otherwise

known as the elasticity modulus, and

usually noted as an uppercase E. Basically,

Young’s modulus describes how easily a

material is compressed or extended

uniaxially. Due to the fact that our model

contains only one material, only one value

for Young’s modulus is given in the

standard model. Unlike the Poisson ratios,

Young’s modulus does not change with

temperature in our models. The unit of

Young’s modulus is usually Pascal (Pa).

The value used in the standard model is

5*1010 Pa, which was chosen as a

reasonable, but rather low, value for

mantle material on Earth. It compares quite

well to the 6.5x1010 Pa that was used by

25

Sandwell and Schubert (Sandwell &

Schubert 1992) in their pure elastic

modelling of the Artemis Corona flexure.

Relaxation time:

The relaxation time of a

viscoelastic material is a parameter that

describes the time over which a

viscoelastic material must be subjected to a

deforming force, in order to react viscously

instead of elastically. The relaxation time,

viscosity and Young’s modulus are

coupled by the equation

Eth = ( 7 )

where h is the viscosity, t the relaxation

time, and E is Young’s modulus. In most

cases, the viscosity is used as the

parameter to describe the viscous

behaviour of a material. In our case, the

relaxation time is used, as this is the

parameter required by the modelling

software ABAQUS. A further advantage of

using the relaxation time is the possibility

to compare the relaxation time of the

material to the time increment used for the

modelling. If the minimum time increment

is larger than the relaxation time, then the

model will treat the material as mainly

elastic. The relaxation time of the mantle

material at 1000 K used in the standard

model is set to 7 years, or roughly 2*108

seconds. The relaxation time of the

effective elastic plate is several ordes of

magnitude larger, as shown by the WLF

plot in figure 19. In the Matlab scripts, the

value is represented by the variable

“relax”.

C1 and C2 coefficients of the Williams-

Landell-Ferry equation

The Williams-Landell-Ferry

equation is used as an approximative way

of obtaining the time-temperature shift

factor described in more detail in chapter

10. The shape and curvature of the

functions graph, shown in figure 19 are

determined by the factors C1 and C2 in

equation 5. Even though C2 is incorporated

into the Matlab scripts as a variable, it was

not intended that we modify this value. For

the standard model it is fixed at a value of

1000. The value of C1 in the standard

model however was designed to be

modified during a model run, and was set

at a starting value of 2. C1 and C2 are

represented by the variable WLF1 and

WLF2 respectively in the scripts.

Thermal conductivity:

The thermal conductivity of a

material describes the rate at which

thermal energy is transported through the

material. This parameter is in our case only

of marginal importance, due to the fact that

our model simply does not run for long

26

enough time spans in order for thermal

conductivity to play much of a role. On the

large scales that are being modelled in this

study, thermal conductive transport is a

slow process, and can largely be ignored.

In order to achieve a temperature

determined density distribution, a full

thermal modelling is required, which

requires a complete set of thermal

properties for the material. The thermal

conductivity is represented by the variable

“conduct” in the scripts and is set to a

value of 3 WK-1m-1 which corresponds to

values used by others in earlier

work(Hofmeister, 1999, Regenauer-Lieb &

Yuen 1998). This value is another

parameter that was not expected to used for

profile fitting.

Specific Heat Capacity:

The specific heat capacity of a

material describes how much thermal

energy a unit of the material absorbs or

emits during the course of changing its

temperature by one thermal unit. This is,

like the thermal conductivity, as parameter

that was included purely for completeness

of the input format for ABAQUS. Together

with the thermal conductivity, and the

initial thermal field, this parameter would

determine the changes in a blocks

temperature due to contact with warmer or

colder blocks. Again, due to the rather

short time scales used in our models,

temperature change is a very small factor,

and is only added for completeness. The

specific heat capacity of the material used

in our models is preset to 1500 JK-1kg-1,

which can be considered a typical heat

capacity of mantle rocks on Earth. The

value is comparable with values from

earlier studies (Regenauer-Lieb & Yuen

1998, 1999, Scott King, 2000).

The table on the following page shows in a

condensed form all the parameters and

geometrical constraints described in this

chapter, together with their values and

adjustability. In total, the number of values

free to adjust comes to 14, whereof 10 or

11 are to be used in fitting the best possible

model profile.

Parameter variable in script starting value unit [SI] ajustable

Vertical model size depth 420’000 m no1

Horizontal model size width 2’400’000 m no1

Size of model elements blocksize 15’000 m yes2

Surface temperature of corona surfacetemp 1000 K yes

Surface temperature of plate platetemp 728 K no

27

Thickness of plate platethick 1 Blocks3 yes

Density of slab subdens 8000 K m-3 yes

Length of slab sublength 150’000 m yes

Gravitational acceleration g 8.87 m s-2 no

Density at plate temperature platedens 3330 kg m-3 yes

Density at mantle temperature mantledens 3250 kg m-3 yes

Poisson ratio of plate platepois 0.25 n/a yes

Poisson ratio of mantle mantlepois 0.35 n/a yes

Young’s modulus Young 5*1010 Pa yes

Relaxation time relax 7 a4 yes

C1 of Williams-Landell-Ferry WLF1 2 n/a yes

C2 of Williams-Landell-Ferry WLF2 1000 n/a no

Thermal conductivity conduct 3 WK-1m-1 yes5

Specific heat capacity specheat 1500 JK-1kg-1 yes5

Table 1: The parameters of the „standard” model

1 These parameters describe the spatial dimensions of the model. They can of course also be modified, but

not without fundamentally changing the nature of the modelled structure

2 The size of the model elements can be modified, but only within the range of block sizes permitted by the

model dimensions. This is clearly a topic of future more refined studies.

3 Exception: Blocks is of course not an SI unit

4 a (years) is also not an SI unit, but is used as a more comfortable unit. For SI units multiply with ~3.1*107

5 These parameters describe the thermal behaviour of the model, and are included only for completeness.

They can of course be adjusted, but are not expected to cause much difference.

15. Finding the best C1 and C2

parameters for the Williams-Landell-

Ferry Equations

The Williams-Landell-Ferry equation,

( )( )02

01 *ttCttCh

-+-

= ( 8 )

that approximates the shift factor described

in chapter 10 is dependent of two

parameters, labelled in most cases as C1

and C2. The value of C2 was decided to be

fixed at a constant 1000. The C1 value is

usually in the regions of 1 to 10. In order to

choose the best fitting C1, the standard

model described in chapter 14 was run 5

times, with C1 values of 2,4,6,8 and 10.

The resulting profiles were then compared

28

to the topographical profile obtain form the

averaging of Artemis profiles. As can be

seen in figure 20, and in the zoomed in

figure 21, the best fit was obtained with a

value of 6 for C1. For further parameter

choosing steps, the WLF parameters will

be set to constant 6 and 1000. Plotted in

Matlab with these parameters for C1 and

C2, the equation describes the curve seen

in figure 19.

16. Fitting the best value for Poisson

ratio of the mantle

The Poisson ratio of the material above the

reference temperature of 1013 K is

assumed to be noticeably higher than the

Poisson ratio of the colder surface material,

and the material cooler than the reference

temperature in general. Poisson ratios must

by definition be between 0.25 and 0.5, with

0.5 being a completely uncompressible

material which is not accepted by the

ABAQUS software. A material with a

Poisson ration below 0.25 would

demonstrate a net gain in volume under

pressure, by expanding laterally by a larger

volume than is shifted by the compression.

Even though some exotic materials with

this property do exist, they can hardly be

isotropic, and are therefore not considered

in this study. Our possible values for the

mantle materials Poisson ratio range from

0.3 to 0.45 in steps of 0.05. These values

most probably contain the correct value

which is expected to be somewhere around

0.33, the typical value for mantle material

on Earth. The standard model described in

chapter 14 was run four times with values

for “mantlepois” of 0.3, 0.35, 0.4 and 0.45

giving us four different profiles. First it

should be noted that the effect on the

profiles was far smaller than expected, and

that all values returned fair fits to the

Artemis Corona profile calculated from the

topography data. As can be seen in figure

22, all four profiles are rather similar, with

the cyan profile (pois.=0.45) matching the

topography best in aspect of slope on the

right hand side of the outer rise flexure.

However the fit of the cyan curve near the

peak of the topography curve is worse than

any others, and the “zigzagging” shape of

the synthetic curve shows buckling in the

model as can be seen in the zoomed in

figure 23. The same applies to a smaller

degree to the yellow curve showing the

synthetic profile with a mantle material

poison value of 0.4. The dark blue curve

produced by setting the value to a low

value of 0.30 shows the worst fit of all,

with an amplitude that is clearly to large,

and a slope curvature that is tighter than

the curvature of the topography. The

overall best fit is presented by the green

curve that shows the synthetic profile with

a mantle Poisson ratio of 0.35. This is near

the expected value of 0.33, but the

29

tendency towards the better slope fit of the

higher values suggests that the actual value

is even slightly higher than 0.35. With a

higher model resolution, the distortion at

the peak of the flexure might be reduced,

giving a better overall fit. For the model

runs performed after this one, the Poisson

ration of the hot mantle material shall be

set to a constant value of 0.35. The also

makes the assumption of a laterally

constant value for temperature and

therefore for the Poisson ratio which is

determined by temperature alone in our

model. This is probably not accurate, in

which case a better fit would be obtainable

by giving the slope area a different state of

underlying mantle material than the peak.

17. Fitting the Poisson ratio of the plate

Consequently, after trying to find the best

fitting value for the Poisson ratio for the

mantle, the next step is finding a best fit

for the Poisson ratio for the plate. The plate

is far colder, and therefore has a lower

Poisson ratio than the underlying material.

In most model, the Poisson value for the

plate is taken to be the common value for

solidified rock, around 0.25. This is also

the value used by the standard model.

Values lower than 0.25 don’t make much

sense, and therefore, the tryout values were

chosen as 0.25, 0.3, 0.35 and 0.4. These

values range up to and beyond the value

determined for Poisson ratio for the

mantle, as shown in the previous chapter.

As can be seen in figure 24, increasing the

Poisson ratio of the plate also increases the

amplitude of the flexure in the outer rise

region. This is to be expected, as the

material is strongly compressed near the

peak, and will spread out more with a

higher Poisson ratio. The best fit here

definitely seems to be the standard value of

0.25. This does not come as a great

surprise, as the plate should behave like

normal solid rock material would. The rim

peak also increases it’s amplitude with

higher mantle Poisson ratios, albeit less

than the outer rise. Even though we are

looking for higher amplitudes in this peak,

the disturbance of the outer rise weighs

heavier. The best fitting value for Poisson

ratio in the mantle material is therefore set

to 0.25 for all further models.

18. Changing the extent of the

downward pull on the plate by changing

the length of the subducted slab

The flexure and consequent upward

bulging of the cold surface material that is

being fitted to the outer rise of the Artemis

profile extracted from the topography data,

is caused by the downward pull of the

subducted material. This material is still

cooler than its surroundings because of the

low heat conductivity value of 3Ws-1m-1

that was chosen for the standard model.

30

Because we use a model made up of one

single material type, the density of the

material is directly coupled to the

temperature of the material. Because the

low conductivity does not allow the

material to quickly increase it’s

temperature in order to reach a thermal

equilibrium, the material is still very much

denser than the surrounding material and

continues to move downwards driven by

gravity. Thereby, it exerts a pull on the

cold material that is left on the surface. The

strength of this pull needed to create a

profile similar to the topographical profile

is quite large, distinctly higher than the

pull exerted by a slab of the same density

as the surface material. At first glance, this

fact alone would prohibit any subduction

process from initiating. In fact the process

of subduction initiation is a wide area of

study and speculation in itself, and shall

not be discussed to any great extent in the

course of this project. In order to simulate

the necessary force to create a suitable

flexure, the temperature of the subducted

material is set to a unnaturally low value

(for Venus) of 400K. The density

associated with such a low temperature is

set to 8000kgm-3 which is in itself an

unnaturally high value. A reason for

artificially increasing the density is to

mimic with a thin slab (15-18km) the same

downward directed body force as a 70km

thick lithosphere. This high density

obviously increases the amount of

downward pull due to gravity, and

therefore provides us with the force

necessary to create our flexure.

Adjusting the amplitude of the flexure, it

would seem, is then a simple matter of

changing the magnitude of the downward

pull. The pull is adjusted by changing the

total mass of the subducted material. For

this, two methods present themselves,

changing the arbitrarily chosen

temperature and density of the subducted

slab, or changing the length of the slab,

thereby adding or removing mass. Our first

approach was to use the second method,

and run our standard model with several

increased slab lengths. The chosen lengths

used for the variable “sublength” in the

model were 150km, 200km, 250km and

300km, with 150km being the value used

in the standard model.

As can be see in figure 25, and the zoomed

in figure 26, the change in subducted mass

doesn’t seem to have very much impact on

the amplitude of the flexure. The right

hand peak depicting the outer rise in the

topographic profile (shown in red) matches

all four synthetic profiles to roughly the

same extent.

The left hand peak of the synthetic profiles

on the contrary shows distinct changes due

to the length of the subducted material.

This peak, representing the inner rim of

Artemis Corona has appeared to be less

31

easily reproduced in our model than the

peak of the outer rise. Strangely enough,

the amplitude of this peak decreases with

increasing strength of the downward pull.

A possible explanation of this behaviour is

the imposed 45° angle at which the slab

subducts. This 45° angle creates a wedge

of mantle material above it that is

separated from the rest of the mantle

material on three sides. In the model, the

left hand peak is caused mostly by the

material displaced by the downward

movement of the slab being pushed

upwards. If the wedge is larger, then most

of the displaced material will be spread out

over the length of the profile, thus creating

a smaller amplitude for the rim peak.

Due to the fact that the amplitude of the

outer rise didn’t change significantly

during these runs, and that creating a

higher inner rim peak seems to require

shorter lengths rather than longer ones, a

new set of model runs was performed with

“sublength” values of 150km, 120km,

100km and 50km. As is visible in figure

27, the shortened lengths of subduction

showed the reverse effect of the above

mentioned lengthening. The shorter the

subducted slab is, the higher the amplitude

of the rim peak becomes. In addition to this

effect, the inner wall of the chasma is

shifted closer to the outer wall with

reduced lengths of subducted material, in

effect narrowing the chasma. The

amplitude of the outer rise peak was again

only marginally affected by changing this

parameter, as was already observed in the

lengthening run. Overall best fit appears to

be achieved with a length of 100km, which

is 50km shorter than originally chosen for

the standard model. The 100km subduction

length profile is plotted in yellow in figure

27 and the zoomed in figures 28 and 29.

The agreement of the yellow curve with

the red topography profile is the best

achieved yet, with a good agreement

between the amplitudes of the natural and

modelled outer rise, and an increasing

correlation of the model with the actual

coronas rim structure. Especially the width

of the chasma is much closer to the

topography than was the case in the

standard model, plotted in dark blue. The

amplitude of the rim is still too small in the

model, and must probably be adjusted by

other means. The optimal amplitude of the

outer rise, and also the optimal width of the

chasma would appear to be slightly below

100km, as the model amplitude of the right

peak is slightly to low with 100km, and the

chasma is still slightly to wide. Further

model runs with more closely spaced

lengths will be performed in the future. In

the following, the standard model will be

set fixed to a subduction length of 100km.

19. Fitting the density of the subducted

slab

32

As changing the length of the slab was

shown, in the previous section, not to have

a significant effect on the size of the bulge,

another method for adjusting the amount of

mass contained in the slab must be used,

namely changing the density of the

subducted material. As we already

accepted the necessity of using unnaturally

high values for this density, changing the

value in this ad hoc way does not

necessarily amount to a more unrealistic

model than without this measure. Of

course, the subducted material in a real

slab is of similar temperature and density

to the plate material at the surface.

Six model runs were performed, all with

rather high densities for the subducted

material. The chosen values were 5’000,

6’000, 7’000, 8’000, 9’000 and 10’000

kgm-3. As can be seen in figure 30,

increasing the density of the subducted

slab had the expected effect of increasing

the amplitude of the surface deformations.

Unlike most other parameters, changing

this value has an almost identical effect on

the outer rise flexure and the rim bulge. Of

all parameters, this value had the largest

effect on the amplitude of both peaks. The

best fit for the flexure zone is obviously

given with a density of 8000kgm-3, which

was incidentally already chosen for the

standard model. The width of the chasma

appears not to differ much between model

runs. For the amplitude of the rim bulge

however, the best fit would actually be for

10’000 kgm-3, and even such a high value

doesn’t generate the amplitudes we are

looking for. Increasing the density of the

subducted material further, in order to fit

the amplitude of the rim would completely

disrupt any fit with the outer rise.

Therefore, overall best fit is accepted as

8000kgm-3, and a different parameter must

provide the necessary uplift for the rim.

8000 is used for further model runs. As

this is the last free parameter that can

sensibly be adjusted, we need to look for a

further option in uplifting the rim peak.

The most obvious method is to introduce

the idea of a mantle plume under the

corona, which pushes the material

upwards. The idea of an active plume

correlates with a theory of corona-genesis

used in earlier studies (Sandwell &

Schubert 1992). Figure 31 depicts the

placement of the “plume” which is

represented by a rectangular block of hot

material placed under the edge of the

corona. The existence of a plume gives us

new variables to adapt, in order to better

our models fit to the actual terrain.

New Parameters due to the introduction

of a mantle plume

20. Plume Temperature

33

Changing the temperature of the plume

material directly changes its density, and

therefore the density contrast with the

cooler surrounding material, This in turn

directly changes the buoyancy of the

plume material, giving us a direct handle

on the upward pushing force applied to the

surface of the model corona. Obviously, in

order to obtain a higher amplitude in the

rim peak, the temperature of the plume

material must be substantially higher than

the surrounding mantle material. Some real

plumes on Earth, however, have been

shown to originate very deep within the

lower mantle (Kiefer and Hager,1991), so

that the very high temperatures needed in

the model can be regarded as roughly

plausible. As can be seen in figure 32, the

mantle potential temperature was elevated

by factors between DT = 100K and an

extreme unreasonable value of DT =

2000K The outer rise peak remained

basically unchanged, as was the intention,

whereas the rim peak becomes slightly

higher with increased temperature and

increased upward pressure. This effect was

expected and intended. The change in

amplitude of the rim peak is nowhere near

the extent expected. It seems that another

parameter of the plume is mainly

responsible for the desired uplift. The

overall best fitting plume temperature was

found to be DT = 500K, which is slightly

in excess of plumes found on Earth..

21. Plume width

The position of the plume was chosen so as

not to disturb the subduction process. The

right edge of the “plume block” is

therefore fixed at a position roughly

100km from the subducting slab. The left

edge however is free to be adjusted. In

figure 32, the position of the edge of the

plume is clearly visible, where the model

profile dips downwards. The largest

possible plume is reached if the plume

extends all the way to the left side of the

model. Therefore, the width of the plume

was chosen as the next parameter to be

adjusted, the results shown in figure 33.

The chosen values were 200, 400, 600 and

800km. The edge of the plume is always

visible in the profiles, with the widest

plume providing the best fit to the

topography. However, the expected change

in amplitude of the rim peak was not, or

only very slightly observable. The

parameter of plume width for further

modelling was set at 800km (i.e. 200km

short of the centre), although probably, a

plume reaching all the way to the centre of

the corona would have produced a better

fit. This is a plausible assumption since

superplume heads on the Earth are found to

have a similar size (Hörnle et al., 2000).

The wide plume models also nicely reflect

34

the central sagging that is visible in the

topography data of the corona.

22. Plume Thickness

The final parameter available to reach

sufficient buoyancy is the thickness of the

plume. By changing this value, we change

the bulk of hot material suspended in the

colder and denser mantle material, and

thereby change the upward pressure. The

parameters used for the model run shown

in figure 34, were thicknesses of one to six

km, where it must be noted that the

thickness parameter actually describes a

variable used in the “makeinp.m” script,

and that the eventual thickness of the

“plume block” is determined by adding

this thickness on to a given level, and also

below this level. Therefore, the actual

thickness of the plume is twice the value

used here. Changing this parameter final

brought about the changes in amplitude to

the rim peak that are necessary for a fit.

The height of the rim peak is increased

with increasing plume thickness. The best

fit however is difficult to determine, as the

deviation from the shape of the peak is still

rather large. The best fit regarding

amplitude would be the maximum value of

6 km, which slightly overshoots the

amplitude of the topography. However, the

peak produced with this parameter is much

wider than the actual rim. It does however

show a fair correlation with the slope of

the rim further to the left. The high

topography to the far left of the profile is

not reproduced in any of the model runs.

This is due to the fact that these high

regions are not directly connected with the

development of the corona, but rather show

overlaying topography. There is an

obvious trade off between plume thickness

and plume excess temperature. Both have

the same effect.

23. Final model

After varying all parameters, and selecting

the best fit for each, a final best fitting

model is obtained. This model will serve as

a preliminary working model for future

fine tuning of parameters. It is not to be

understood as a satisfactory fit. This model

as shown in figure 35 correlates fairly with

the topography profile extracted from

Artemis Corona. As the whole modelling

was done as an axissymmetric model, a 3D

extraction can be constructed as is shown

in figure 36. This compares roughly to the

Artemis topography image shown in

comparison in figure 37. The final

parameters and given in the following

table:

35

Parameter Unit Range tried Value chosen

WLF1 n/a 2 ,4 ,6 ,8 , 10

Poisson mantle n/a 0.3, 0.35, 0.4, 0.45 0.35

Poisson plate n/a 0.25, 0.3, 0.35, 0.4 0.25

Subduction length km 300, 250, 200, 150,

120, 100, 50 100

Subduction density kgm-3 5000, 6000, 7000,

8000, 9000, 10000 8000

Plume temperature K DT = 100 , 500, 1000,

1500, 2000 DT = 500

Plume width km 200, 400, 600, 800 800 +

Plume thickness km 2, 4, 6, 8, 10, 12 12

Unfortunately, due to time limitations, it

was impossible to run further models in

order to fine tune the parameters to a

greater precision. Further work on the

model would definitely produce better

results, and therefore allow more detailed

conclusions about the actual conditions in

the upper mantle regions of Venus. In

particular, modelling the Corona with more

closely spaced parameter sets, is bound to

produce better results. A similar model,

that incorporates the plume from the

beginning, or even uses only the plume

might result in a more realistic

representation of the mantle structure.

Further, modelling this structure under

consideration of plastic and failure criteria

would also reveal a lot more information,

especially concerning the shape of the

Chasma. Applying the same modelling

principles to other coronae might shed

light on the question of whether Artemis is

really just the largest structure of it’s kind,

or completely of different origin.

Modelling from the beginning of plume

lithosphere impact to foundering of the

lithosphere to subduction initiation to

corona formation would definitely be a

promising project for the future.

Conclusions

First part:

The Mexican Hat Wavelet analysis

method developed in this project has

fulfilled and partly surpassed our

expectations. It could be shown that the

outer rise near a known subduction zone on

Earth is of a constant typical wavelength

36

even under complex geometrical

configurations like in the Aleutian. This

strongly suggest it’s origin in the elastic

properties of the lithosphere and it is

difficult to conceive a significant role in

the mantle flow –slab feedback

mechanism. Also, similarly, constant

wavelengths were detected along the outer

rise of Artemis Corona on Venus,

suggesting it’s relation to subduction zones

or other zones of strong lithosphere flexure

on Earth. The analysis method shows a

great potential for further development.

Among other possibilities, a more detailed

use of the method during a second run with

tighter spaced wavelength could determine

the actual wavelength of the flexure to a

higher accuracy, as well as determine the

degree of variance in wavelength along the

structure. Studies with other wavelet

shapes might be applied to structures of

different origins. A most promising idea

for further analysis would be the

application of the method to gravity data.

Due to the generally smoother nature of

gravity data in comparison to topography,

the results of the wavelet analysis should

be of significantly higher quality. Joint

wavelet analysis of topography and gravity

data over the same region would most

probably lead to a powerful tool for the

mechanical analysis of lithosphere

properties.

Second part:

In spite of the preliminary,

exploratory, nature of the potential for

describing the lithosphere mantle system in

terms of a visco-elastic Williams-Landell

Ferry model our runs undisputedly show a

large similarity to the topography profile

extracted from the Magellan data over

Artemis Corona. The best parallels are

found in the shape and amplitude of the

outer rise, as well as in the width of the

chasma. The rim of the corona is less well

modelled. Due to the short time available

for this project, many parameters could

only be determined very roughly. The fact

that “a fair” fit could be achieved by such

widely spaced parameter selections is a

strong argument for the hypothesis of

Corona origin in circular subduction zones.

The shortcoming of the model are mainly

found in the depth of the chasma, which is

either due to the models lack of plastic and

failure criteria for the material, or maybe

also due to the fact that the radar survey of

the Magellan probe could not penetrate to

the full depth of the chasma, or is simply

attributed to a systematic error our choice

of permissible parameter ranges (e.g. the

mantle viscosity).

Moreover, the need for significant

upwelling in order to achieve the necessary

surface amplitudes inside the Corona and

at its rim seem like a brute force approach.

The non-failing (plastic) nature of the

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surface of the model, however, places an

additional downward pull on the rim from

the subduction zone, which could be much

lessened by simulating a failing material.

This would probably reduce the need for

the buoyant vigour of the hot plume. That

any uplift is necessary however seems to

be an argument for still active support

under the Corona.

The unnaturally high density applied to the

subducted slab is due to the models

preference for short slabs by applying the

same downwards directed body force. A

short slab is necessary to avoid shielding

of the rim region. Again, failing surface

could probably decrease the necessity of

such implausibly high values. Further

development of the model must therefore

include plastic properties, and would

probably better be modelled in a fully

dynamic way. A model that starts with the

plume and generates a Corona from the

beginning would give further insight and

verify if the selected mechanism is actually

capable of creating these structures. A

similar model should also be applied to

smaller Venusian coronae, in order to

determine if Artemis is actually a typical

example of these strange structures.

Overall, the model lends further credibility

to the hypothesis of plume and retreating

subduction proposed by Sandwell and

Schubert (Sandwell and Schubert, 1992).

We have tested the hypothesis in a

dynamic setup and find additional

observables such as inner rim topography

and chasma wavelength to be consistent

with the intricate link of corona structure

and plume activity

Acknowledgments

Thanks above all go to Profs. Regenauer-

Lieb and Giardini, for giving me the

opportunity to work on this project, and

especially to Klaus Regenauer and

Gabrielle Morra for their tireless assistance

and patience. Further thanks go to Heinrich

Horstmeyer for his assistance where my

Matlab programming abilities fell short.

Many thanks also go to my parents, B. and

D. Mettier, for their moral and financial

support not only during the work on this

diploma thesis, but also during the whole

27 Years of my education. Finally, and

maybe most important, my thanks go to

Monika Bitter, for being my support,

inspiration and motivation, and basically,

just for being there.

.

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Appendix A : Images

Figure 1: Labelled topographic map of Venus, compiled from Magellan Radar data. The Map shows the two main highland areas, Aphrodite Terra and Ishtar Terra, as well as the vast flat lowlands, that make up most of Venus’ surface area. This image is a courtesy of NASA.

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Figure 2: Alpha regio with Eve. Alpha Regio is one of the smaller highland areas one Venus, named due to it being the first surface structure that was identified by radar surveys of the planet. The Ovoid structure in the lower left corner of the image is named Eve (all structures are name after women on Venus, and Eve was the biblical first woman, hence the name for the first structure identified). Nearly exactly in the centre of Eve is a small light patch of highly radar reflective material. This feature is defines the prime meridian, and therefore the whole of the geographic grid on Venus. This 3D image was constructed from Magellan altimetric data. The image is a courtesy of NASA.

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Figure 3: Elevation Histograms of Venus and Earth. The distribution of terrain elevations over the two planets reflects their differences in surface appearance. The bimodal distribution of Earths elevations reflects the ocean floors and continents, the two dominant terrain types on Earth. Venus has a unimodal distribution, with the maximum near the average elevation or datum. The slight biasing of Venus elevation histogram towards higher regions is because highlands are slightly more common than lowlands on Venus. The overall vertical relief is larger on Earth. Image is taken from (Cattermole 1994).

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Figure 4: The Magellan spacecraft, after release from the Space shuttles cargo bay. Magellan was the first interplanetary craft to be launched from the shuttle and returned a plethora of data of hitherto unknown quality about the planet Venus. Image courtesy of NASA.

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Figure 5: The complete topographical dataset for Earth used in this project. The image was compiled from the Smith & Sandwell topography and Bathymetry dataset (Smith & Sandwell, 1997) using the Matlab routines “mygrid_sand.m” and “makeearth.m”.

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Figure 6: Topographical map of the Aleutian island arc. The trench caused by subduction is well visible, the upward flexure of the outer rise is also visible, albeit less obvious. This image is constructed from the Sandwell bathymetry dataset combined with the GTOPO30 topography data (Smith and Sandwell, 1997).

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Figure 7: Topographical map of Artemis Corona, Venus. The chasma is well visible, as is the outer rise of this ring shaped structure. The data for this map is the Magellan Venus topography dataset (Courtesy of NASA). The image was created in Matlab. Projection is Mercator, axes are labelling distances in km.

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Figure 8: Average profile of the southeastern quadrant of Artemis Corona. The profile was created by averaging 19 profiles extracted from the Magellan topography dataset. The high similarity of the 19 extracted profiles in the rim and outer rise region underlines the circular nature of the Corona. In addition, creating an average profile functions to a certain degree as a smoothing procedure for short wavelength topography, which is not coupled to the coronas nature. The profile was extracted and averaged by the Matlab script “getprofile.m”(Appendix B). The centre of the corona is located to the left of the profile. The Chasma and the outer rise are very obvious.

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Figure 9: The standard 2D Mexican Hat Wavelet, as produced by the Matlab function “mexihat“. The wavelet is the negative second derivative of a gaussian pulse (bell curve). In this image, it is clear that the wavelet integrates to zero, and is symmetric in respect to the y-axis.

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Figure 10: The Mexican Hat Wavelet adapted for 3D. This surface plot was achieved by rotating the Wavelet function shown in figure 4 around the y-axis. This is the template for all wavelets used in the analysis of the topography data of this project.

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Figure 11: Our definition of Wavelength as compared to the real wavelength. The red line shows the standard Mexican Hat Wavelet, as also seen in figure 4. The blue line represents a cosine function of the same wavelength as the wavelet. The inner arrows define our “wavelength” as used in the topography analyses. The outer arrows signify the true wavelength of the cosine function, which in this case depicts a hypothetical topography structure. Our definition was chosen for two main reasons. Firstly, as is obvious in this figure, it is rather difficult to identify the true wavelength of the wavelet as there is no telltale point as in the cosine function. Secondly, topographic structures often only display the positive half cycle, making it hard to judge their true wavelengths. As the whole point of identifying wavelengths of topography structures is to compare them with each other, this somewhat exotic definition is valid as long as it is applied to all examples.

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Figure 12: Schematic illustration of a typical subduction zone. The lower example shows the Japanese trench, but could just as well apply to the Aleutian trench. Not shown in this simplified figure is the typical outer rise that is created through lithosphere flexure seawards of the trench. Image is taken from Press and Siever, Understanding Earth, 1998.

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Figure 13: Deflection from Vertical (DFV) Map of Alaska and the Aleutian island chain. The colouring of the image indicates tilt of terrain in North/South Direction. The Aleutian trench and especially the outer rise are extremely well visible in this type of map. Nonetheless, determining the wavelength of the structure is rather difficult, due to the asymptotic recline of the outer rise.

Image courtesy of NOAA, http://www. ngs.noaa.gov/GEOID/IMAGE96/

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Figure 14: Comparison of the topographical data of the Aleutian trench region, before and after the smoothing process (“Imageflat.m“) is applied. Long wavelength structures are enhanced by the smoothing process, whereas short wavelength disturbances are slightly dampened. The main point of this procedure is to reduce the data amount that is used as input for the analysis scripts. As can be seen in these maps, this smoothing method does not reduce the quality of the data used for detecting lithosphere flexure structures.

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Figure 15: Final results of the Mexican Hat Wavelet analysis of the Aleutian trench. The top image shows the original topography data used as input for the analysis scripts. The colorbar depicts altitudes in km. The lower image is the final plot of the analysis process. Colouring depicts the dominant wavelengths at each point on the map. The colorbar shows the wavelengths in km (our wavelength definition).The trench and outer rise can easily be identified as the continuous bands of orange, blue and yellow. The yellow band identifies the outer rise, with a wavelength of roughly 350km. In both images, axes depict distances in km.

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Figure 16: Profile across the Aleutian island arc and trench. Arrows indicate the trench and outer rise. Wavelengths can be roughly judged as ~350km for the outer rise, and ~150km for the trench. The profile was extracted with the “getprofile.m” script (Appendix B), from the Smith and Sandwell topography data (Sandwell & Smith, 1997). X-axis shows distance in km.

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Figure 17: Final results of the Mexican hat Wavelet analysis of Artemis Corona. The top image shows the topography of the region, as in figure 2. The bottom image shows the same region colour code for dominant wavelengths, comparable to the Aleutian results in figure 10. The trench is visible as a continuous band of darker blue, representing a wavelength of around 250-300km. The outer rise is shown as a yellow/green band giving a wavelength of 650-750 km. In general, these values are double those detected for the Aleutian flexure zone. Note the wider zone of boundary effects (dark blue frame) than in the Aleutian analysis. This stems from the larger wavelengths, and therefore larger hats that were necessary.

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Figure 18: The „standard“ model in its starting state. The colours depict temperature, as seen in the colorbar at the top left.

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Figure 19: The graphs described by the Williams-Landell-Ferry equation. Basically, it describes the change to the relaxation time of a material due to temperature. The glass transition temperature is used as the reference temperature t0. At the point of t0, the log of the shift is by definition 1, giving a shift of 0. Therefore, the relaxation time given in the variable “relax” describes the relaxation time at the temperature that is used as the reference temperature, here 1000 K. Upwards of the reference temperature, the shift changes only minimally, which is correct, as hotter material should stay mainly viscously dominated. Below the reference temperature, the shift becomes large rather quickly, causing the relaxation time of the material to increase dramatically. As the relaxation time rises to a value large than the time increment used in the model, the material starts behaving mainly elastically, which is what we need for the cooler material of the plate. The value of C1 that is changed in this figure determines mainly how sharply the curve turns below the reference temperature.

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Figure 20: Surface profiles modelled by changing the parameter WLF1 (C1). The right peak, which represents the outer rise, is better approximated than the left peak, which represents the rim of the corona. The central trench of the model is far deeper than the actual Artemis Chasma, and has been clipped to a maximum depth of 1000m. This difference is mainly caused by the model not using any plastic or failure criteria for the material.

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Figure 21: Zoomed in view of the outer rise peak from figure 20.

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Figure 22: Profiles extracted from the models run with different values for the Poisson ratio of the mantle.

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Figure 23: Zoomed in view of the outer rim peak from figure 22

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Figure 24: Profiles extracted from models run with varying values of Poisson ratio for the surface plate material.

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Figure 25: Profiles extracted from the models run with varying lengths of subducted material. Note the tendency of the left peak (rim) to loose in amplitude, as the plate gets longer, and therefore heavier. This is probably due to a shielding effect from the slab, preventing the displaced material from creating the bulge.

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Figure 26: Zoomed in view of the outer rim peak from figure 25. As is shown in this figure, the change of length of the subducted material, and hence the change in mass does not seem to have a very large effect on the amplitude of the outer rise peak. Obviously, increasing the mass of the slab should place more strain of the plate, and increase the amplitude. However, some mechanism seems to be countering this effect.

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Figure 27: Profiles extracted from models run with decreasing lengths of subducting material. As comparison to figure 25, the length of 150 km is used in both model runs. Note the impact of reducing the length to very small values (50km), as compared to the low impact of changing form 150 down to 100km length. Strangely enough, shortening the slab increases the amplitude of the rim peak, where lengthening reduced it. The best fitting length seems to be around 100km.

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Figure 28: Zoomed in view of the outer rise peak of figure 27. The worst fit is given by the 50km length, whereas all other lengths return fair fits.

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Figure 29: Zoomed in view of the rim peak and chasma from figure 27

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Figure 30: Profiles extracted from models run with varying values of density for the subducted slab material. Unlike most parameters, the effect is roughly the same on both peaks.

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Figure 31: The need for a different mechanism to provide the uplift under the rim peak caused us to introduce a „plume“ to the model. The above image shows the first location of the plume, which is very simply depicted by a rectangular box of hot material placed under the surface. Using this plume gives us three new parameters to modify, the width, height and density of the plume material.

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Figure 32: Profiles extracted from the models run with a newly introduced “plume”, and varying values for plume temperature. The dip at about 40 on the x-axis shows the effect of the plume as compare to without any underlying hot material. The outer rise peak is generally unaffected by the plume, as was expected, and the rim peak shows a very promising increase in amplitude. Best overall fit is given by the DT=500K curve.

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Figure 33: Profiles extracted from the models run with variations of the width of the plume. The amplitudes of the peaks remained largely unchanged by this process, but the edge of the plume is very well visible as the dip in the profile.

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Figure 34: The profiles extracted from the model during modification of the last available variable, the thickness of the plume. The values given in the legend must be understood as half the thickness of the plume, due to a programming problem.

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Figure 35: Comparison of the true Artemis topography, shown in red, and the overall best fitting model. The fit of this model in the outer rise peak is less than previous runs. The fit however at the rim peak is quite good. The overall fit must be a compromise between the fits at the two peaks. With more closely spaced parameters, a better fit would most probably be obtainable.

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Figure 36: 3 dimensional view of the axissymmetric model. Comparable to the 3D surface plot of the Artemis topography data shown in figure 37. The trench is clearly visible, with the yellow/green bands representing the outer rise and the rim or the Corona. The northwestern Quadrant was purposely left away to mimic the shape of Artemis. The green dip to the centre of the model is due to the plume not being modelled all the way to the model rim.

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Figure 37: Topographic data of the actual Artemis Corona. This image can be compared to the model shown above. Especially trench, and outer rise are fairly well correlated between these two images.

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Appendix B: Matlab Scripts This appendix contains the Matlab scripts used for this projects. Unless noted otherwise, the scripts are developed and written by Ralph Mettier, for “the Mathworks” math software package “Matlab version 6.1”. The scripts are presented in alphabetical order with the text colouring adopted from the original Matlab editor.

1. Artprof.m

% Artprof.m is a script that extracts a profile from the % % Artemis Corona topographic data % clear close all clc load Artkm imagesc(Artkm) axis equal tight [x,y]=ginput(3) d=sqrt(((x(2)-x(3))^2)+((y(2)-y(3))^2)); Artem=imcrop(Artemis,[x(1)-d,y(1)-d,2*d,2*d]); figure imagesc(Artem) axis equal tight save Artem Artem Xc=x(1); Yc=y(1); [x,y]=size(Artem) Artem1=Artem; profiles=zeros(19,floor(x/2)); for i=1:19 angle=2*pi*((i-1)*5)/360 for j=1:floor(x/2) [xtemp,ytemp]=(pol2cart(angle,j)); xtemp=round(xtemp+x/2); ytemp=round(ytemp+y/2); profiles(i,j)=Artem(xtemp,ytemp); Artem1(xtemp,ytemp)=30000; end plot(profiles(i,:)) hold on end

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figure imagesc(Artem1) profs=sum(profiles,1)/19 dist=length(profs); prof=interp1(profs,linspace(0,length(profs),dist)); plot(0:dist-1,prof) figure imagesc(Artem1) caxis([10000 16000]) colorbar

Artprof.m extracts several radially arranged profiles from a input topographical dataset of Artemis Corona (Artkm.mat) with 1km resolution. The script first produces a map view of the topography data and expects three data points to be provided by mouseclicks. Of these three points, one should be the approximate centre of the circular structure, the other two points should be situated on the perimeter of the Corona, in order to allow the script to determine the position and diameter of the strucute. The centre is then set as the new origin of a polar coordinate system, and profiles are extracted reaching from the centre to twice the structures radius, in 5 degree spacings. The double radius length was chosen in order to position the region of interest, namely the rim, chasma and outer rise, near the middle of the profile. The coordinates of each profile sample are calculated in the polar coordinate system, and then transformed back to cartesian coordinates and extracted from the original data, In a copy of the data matrix, the profile coordinates are coloured red, in order to give the user a control feature as to where the profiles are extracted. Finally, the profiles are averaged in order to produce a final “typical” profile of the structure. This profile is then used as a comparison for the modelled profiles.

2. choose_WLF1.m clear load Profmatrix [x,y]=size(Profmatrix); hold on color=['b' 'g' 'y' 'c' 'k']; for i=8:8:40 Profmatrix(i,:)=Profmatrix(i,:)-Profmatrix(i,end); a=(Profmatrix(i,:)<=-1000); Profmatrix(i,a)=-1000; plot(Profmatrix(i,:),color(i/8)) end load Artprof prof=prof-prof(end); prof=interp1(prof,linspace(0,length(prof),y)); prof=prof(11:end); plot(prof,'r')

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grid on legend('WLF1=2','WLF1=4','WLF1=6','WLF1=8','WLF1=10','topo') The “choose*.m” scripts are basically different versions of the same script. As an example, choose_WLF1.m is shown here. The “choose” scripts are designed to display the profiles of the different models in after a run changing one parameter has been completed. It loads the Profmatrix.mat data file that is output by filconvert.m, and plots the selected profiles in different colours. In order to adjust the script for different variables, the length of the for loop must be modified for the number of models run, and the legend must be chaged to show the correct labels. 3. filconvert.m

fin=fopen('test.fil'); foutorg=fopen('test.fle','w'); fout=foutorg; ch=[]; str=[]; ender=1; a=0; while ender ch=fscanf(fin,'%c',1); if strcmp(ch,'')==1 ch=['?']; end switch double(ch) case 10, ch=[]; str=[str ch]; case 68, plustest=fscanf(fin,'%c',1); if strcmp(plustest,'+')==1 ch=['e+']; elseif strcmp(plustest,'-')==1 ch=['e-']; elseif strcmp(plustest,' ')==1 ch=[' ']; end str=[str ch]; case 63, ender=0; case 73, pseud=fscanf(fin,'%c',2); str=[str ' ']; case 42, str=[' ' str]; f=findstr(str,'SURP'); e=findstr(str,'2 2001');

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if f

fprintf(fout,[str '\n']); eval(['fout=fopen(''profile' num2str(a) '.fle'',''w'')']) a=a+1; end if e fout=foutorg; end fprintf(fout,[str '\n']); str=[]; otherwise, str=[str ch]; end end disp('end of file reached') %figure %hold on %load ../Artprof load params load ../Profmatrix; for i=1:a-1 eval(['! paste p.txt profile' num2str(i) '.fle >profile' num2str(i) '.prf']) eval(['data=load(''profile' num2str(i) '.prf'');']) profile=data(:,5); profsize=length(profile)*blocksize; modprof=interp1(linspace(0,profsize,length(profile)),profile,linspace(0,profsize,profsize/10000)); if length(modprof)>240 modprof=modprof(1:240); end Profmatrix=[Profmatrix;modprof]; %plot(modprof) %drawnow end save ../Profmatrix Profmatrix ! rm *.fle

filconvert is a key function to this project. All models were computed in ABAQUS, which is capable of exporting various data to ascii output files. Exporting the surface profiles of the models yielded the *.fil file format, which is rather cryptic. In order to further process the data with Matlab, a routine that converts the .fil file to a more readable format was necessary. filconvert reads the .fil file and outputs the contained profiles each to it’s own file, The non-profile part of the .fil file is output to a dummy file. The script also creates the “Profmatrix.mat” which is the main input for the subsequent “choose” script. The filconvert script contains many Unix shell commends, identifiable by the exclamation

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marks at the front of the line. This was necessary in order to manipulated the many files that were created, and also in order to delete the temporary and dummy files used. If this script is used in a MS-Windows based Matlab, these instructions will have to be modified. 4. fixmapvenus.m clear all close all clc load Venus data=Venus; [Nx,Ny]=size(Venus); for i=1:Nx line=Venus(i,:); IX=find(line~=0); data(i,:)=interp1(IX,line(IX),1:Ny); end figure subplot(1,2,1) imagesc(Venus) caxis([10000 14000]) subplot(1,2,2) imagesc(data) caxis([10000 14000]) figure imagesc(data-Venus) caxis([10000 14000]) Fixmapvenus.m is a script that was used to interpolate over the areas where the Magellan data shows gaps. These gaps represent areas where no data was recorded, and show up as zeros in the original dataset. As these zero value areas create boundary effects during the wavelet analysis, they were interpolated over. Because of the size of the dataset, the standard 2D interpolation routines integrated in Matlab were unable to work. Therefor, the fixmapvenus.m script runs through each line of the dataset and applies a 1D interpolation routine. Because the gaps follow the orbits of the Magellan craft, they are much longer than wide, and therefore the 1D interpolation should be valid.

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5. Imageflat.m function Data=Imageflat(data,tilesize); a=tilesize; % Plot the original data figure subplot(1,2,1) imagesc(data) axis equal tight title('Before') % crop the size of the data to a whole multiple of the tile size [x,y]=size(data); data=imcrop(data,[1,1,(floor(y/a)*a)-1,(floor(x/a)*a)-1]); % create the tiles [x,y]=size(data); temp=0:a:x; xcounter=temp(2:length(temp)); clear temp temp=0:a:y; ycounter=temp(2:length(temp)); clear temp %Sum and average the values foreach tile for i=1:length(ycounter) for j=1:length(xcounter) Data(j,i)=(sum(sum(data((xcounter(j)-a+1):(xcounter(j)),(ycounter(i)-a+1):ycounter(i)))))/(a^2); end end % Plot the flattened data subplot(1,2,2) imagesc(Data) axis equal tight title('After') Imageflat.m is another much used script, which basically performs a square average smoothing to any input data matrix. The main reason for this script is to reduce the overwhelming matrix sizes associated with the topography data used. With 1km resolution data, studying 2000-3000 km size structure rends enourmous matrices which strain the capabilities of just about any interpolation or correlation routine. As the wavelengths we were looking for are much larger than 1km, it is possible to reduce the amount of input data without disturbing the information content. The script expects an input variable giving the side length of the averaged squares. It then divides the input matrix into as many squares of this size as possible, and calculates the average of the values inside each square. The results are then plotted as the reduced data map. A further advantage of this method, even though this was never planned, is the low pass filter effect. Structures with

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short wavelengths are suppressed, allowing long wavelength structures to become better visible. The script also displays a before/after comparison, which allows the user to judge the quality loss in the reduced data map. 6. makeearth.m clear close all clc Earth=[]; x=1:114; y=1:1600; [X2,Y2]=meshgrid(x,y); for i=5:5:360 [data,vlat,vlon] = mygrid_sand([-72 72 i (i+5)]); [x,y]=size(data); X1=1:x; Y1=1:y; Data=griddata(vlon,vlat,data,X2,Y2); Earth=[Earth Data]; end pcolor(Earth); Makeearth.m is a short script written in order to extract a global topography map from the Sandwell topo8_1.img data file. The data file is normally read by the mygrid_sand.m script provided together with the data. However, the mygrid_sand.m script can only handle a limited amount of data. Therefor, the makeearth.m script calls the mygrid_sand.m script to extract five degree swaths from the data file, and then assembles them to a complete map. The topography data is then displayed in map format. 7. makeinp.m

% makeinp.m generates a standardized input file for Abaqus, representing a profile through Artemis corona % point of the script is to enable running several models with different starting parameters from a single command % Author: Ralph Mettier 25.06.02 % Generate the Nodes and Elements from width, depth and element size function []=makeinp(modelnumber,depth,width,blocksize,SurfaceTemp,platetemp,platethick,subdens,sublength,g,platedens,mantledens,platepois,mantlepois,Young,relax,WLF1,WLF2,conduct,specheat)

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% depth, width, blocksize, sublength in [m]; surfacetemp, platetemp in [K]; platethick in [blocks]; g in [m/s^2]

modelnumber eval(['!mkdir model' num2str(modelnumber)]); eval(['cd model' num2str(modelnumber)]); eval(['!cp ../test.inp .']); eval(['!cp ../filconvert.m .']); eval(['!cp ../Output.abq .']); eval(['!cp ../p.txt .']); pwd foutN=fopen('Nodes.abq','w'); foutE=fopen('Blockelements.abq','w'); x=(0:blocksize:width)'; y=(0:blocksize:depth)'; NodesX=zeros(length(x)*length(y),1); NodesY=zeros(length(x)*length(y),1); NodesIndex=zeros(length(x)*length(y),1); NodesX=repmat(x,length(y),1); NodesY=reshape(repmat(y',length(x),1),length(x)*length(y),1); NodesIndex=(1:length(x)*length(y))'; fprintf(foutN,'*NODE \n') fprintf(foutN,'%9d, %9d, %9d \n',[NodesIndex NodesX NodesY]') fprintf(foutN,'**') fclose(foutN) ElemIndex=zeros((length(x)-1)*(length(y)-1),1); ElemLU=[]; ElemRU=[]; ElemRO=[]; ElemLO=[]; ElemIndex=(1:length(ElemIndex))'; for i=1:length(y)-1 ElemLU=[ElemLU; NodesIndex((i-1)*length(x)+1:(i-1)*length(x)+length(x)-1)]; ElemRU=[ElemRU; NodesIndex((i-1)*length(x)+2:(i-1)*length(x)+length(x))]; ElemRO=[ElemRO; NodesIndex((i)*length(x)+2:(i)*length(x)+length(x))]; ElemLO=[ElemLO; NodesIndex((i)*length(x)+1:(i)*length(x)+length(x)-1)]; end fprintf(foutE,'*ELEMENT, ELSET=BLOCK, TYPE=CAX4 \n') fprintf(foutE,'%9d, %9d, %9d, %9d, %9d \n',[ElemIndex ElemLU ElemRU ElemRO ElemLO]') fprintf(foutE,'**') fclose(foutE)

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%testinp % a short script to test if all elements are described

% SURP = Elementset that is written to an ASCII file, and can be read using filconvert.m SURPout=fopen('SURP.abq','w'); SURPmin=NodesIndex(end-length(x)+1); SURPmax=NodesIndex(end); Step=1; fprintf(SURPout,'*NSET,NSET=SURP,GENERATE \n'); fprintf(SURPout,'%5d, %5d, %3d \n',[SURPmin SURPmax Step]'); fprintf(SURPout,'**'); fclose(SURPout) % Box contains all Floor and Sides Boundary Conditions Boxout=fopen('Box.abq','w'); FloorIndex=NodesIndex(1:length(x)); FloorDir=2*(ones(size(FloorIndex))); FloorMove=zeros(size(FloorIndex)); fprintf(Boxout,'** floor \n'); fprintf(Boxout,'*BOUNDARY, OP=MOD \n'); fprintf(Boxout,'%4d, %2d,,%9d \n',[FloorIndex FloorDir FloorMove]'); fprintf(Boxout,'** \n'); EleMatrix=flipud((reshape(ElemIndex,length(x)-1,length(y)-1))'); NodesMatrix=flipud((reshape(NodesIndex,length(x),length(y)))'); SidesIndex=sort([flipud(NodesMatrix(:,1)); flipud(NodesMatrix(:,length(x)))]); SidesDir=ones(size(SidesIndex)); SidesMove=zeros(size(SidesIndex)); fprintf(Boxout,'** sides \n'); fprintf(Boxout,'*BOUNDARY, OP=NEW \n'); fprintf(Boxout,'%4d, %2d,,%9d \n',[SidesIndex SidesDir SidesMove]'); fprintf(Boxout,'**'); fclose(Boxout) % Initial Temperature field is generated Tempout=fopen('Temp.abq','w'); TempMatrix=zeros(size(NodesMatrix)); gradient=3.5; TempCurve=round(linspace(SurfaceTemp,depth/1000*gradient+SurfaceTemp,length(y)))'; TempMatrix=repmat(TempCurve,1,length(x)); % additional temperatur data, for non-smooth fields %

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platerim=length(x)-round(length(x)/2); TempMatrix(1:platethick,1:platerim)=platetemp; for i=1:floor(sublength/blocksize) TempMatrix(platethick+i-1:platethick+i,platerim+i:platerim+i+1)=400; end TempIndex=flipud(reshape(TempMatrix',size(NodesIndex))); fprintf(Tempout,'*INITIAL CONDITIONS, TYPE=TEMPERATURE \n'); fprintf(Tempout,'%5d, %5d \n',[NodesIndex TempIndex]'); fprintf(Tempout,'**'); fclose(Tempout) % The two parts of the Gravity data are written, one has to be inserted before the step starts, one into the step Gravout1=fopen('GravSet.abq','w'); Gravout2=fopen('Gravity.abq','w'); fprintf(Gravout1,'** Gravity \n'); fprintf(Gravout1,'*ELSET, GENERATE, ELSET=GRAVITY \n'); str=[' ' num2str(min(ElemIndex)) ', ' num2str(max(ElemIndex)) ', ' '1']; fprintf(Gravout1,[str '\n']); fprintf(Gravout2,'*DLOAD, OP=NEW \n'); str=['GRAVITY, GRAV, ' num2str(g) ', 0., -1., 0.']; fprintf(Gravout2,[str '\n']); fprintf(Gravout2,'**'); fclose(Gravout1) fclose(Gravout2) Contout=fopen('Contact.abq','w'); ContIndex=ElemIndex(end-length(x)+2:end); fprintf(Contout,'*SURFACE,NAME=M1,TYPE=ELEMENT \n'); fprintf(Contout,'%5d, S3 \n',ContIndex); fprintf(Contout,'*CONTACT PAIR,INTERACTION=I1,SMOOTH=0.45 \n'); fprintf(Contout,' M1, M1 \n'); fprintf(Contout,'*SURFACE INTERACTION,NAME=I1 \n'); fprintf(Contout,'*FRICTION,TAUMAX=1000.,SLIP TOLERANCE=0.02 \n'); fprintf(Contout,'0.001, \n'); fclose(Contout); Matout=fopen('Material.abq','w'); mintemp=floor(platetemp/100)*100; maxtemp=max(max(TempMatrix))+1000; Youngnum=floor(Young/10^8)/100;

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relaxnum=(floor(relax*3*10^7)/(10^6))/100; fprintf(Matout,'*SOLID SECTION,MATERIAL=STUFF,ELSET=BLOCK \n'); fprintf(Matout,'*MATERIAL,NAME=STUFF \n'); fprintf(Matout,'*DENSITY \n'); fprintf(Matout,' %4d., %4d \n',[subdens 400]); fprintf(Matout,' %4d., %4d \n',[platedens+50 mintemp]); fprintf(Matout,' %4d., %4d \n',[platedens SurfaceTemp]); fprintf(Matout,' %4d., %4d \n',[mantledens+50 SurfaceTemp+1]); fprintf(Matout,' %4d., %4d \n',[mantledens-50 maxtemp]); fprintf(Matout,'*ELASTIC,TYPE=ISO,MODULI=INSTANTANEOUS \n'); fprintf(Matout,' %3.2fE+10, %3.2f, %5d \n',[Youngnum platepois mintemp]); fprintf(Matout,' %3.2fE+10, %3.2f, %5d \n',[Youngnum platepois+0.01 SurfaceTemp]); fprintf(Matout,' %3.2fE+10, %3.2f, %5d \n',[Youngnum mantlepois-0.01 SurfaceTemp+1]); fprintf(Matout,' %3.2fE+10, %3.2f, %5d \n',[Youngnum mantlepois maxtemp]); fprintf(Matout,'*EXPANSION,ZERO=0.,TYPE=ISO \n'); fprintf(Matout,' 3.1E-5, \n'); fprintf(Matout,'*VISCOELASTIC,TIME=PRONY \n'); fprintf(Matout,' 1., 0., %3.2fE+8 \n',relaxnum); fprintf(Matout,'*TRS,DEFINITION=WLF \n'); fprintf(Matout,'%5.1f,%1d,%4d \n',[SurfaceTemp WLF1 WLF2]); fprintf(Matout,'*CONDUCTIVITY \n'); fprintf(Matout,'%2.1f \n',conduct); fprintf(Matout,'*SPECIFIC HEAT \n'); fprintf(Matout,'%5.1f \n',specheat); fclose(Matout) save params modelnumber depth width blocksize SurfaceTemp platetemp platethick subdens sublength g platedens mantledens platepois mantlepois Young relax WLF1 WLF2 conduct specheat; pwd eval(['!abaqus interactive job=test']) disp('model done, extracting profiles') filconvert cd .. disp(['all done with model' num2str(modelnumber)])

makeinp.m is probably the most important script in this projects. The ABAQUS modelling software needs the model parameters to be provided in a input text file. For the size of our model, this file reaches considerable length, and is also rather difficult to read. The input files are usually created by preprocessing software like PATRAN or similar programs. However, changing a singular parameter before running a model would also require rerunning the preprocessing step again. This is very impracticable for our purposes. Thererfore, makeinp.m creates several *.abq text files which can then be imported into the input fil via the *INCLUDE option. This greatly simplifies the work of

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creating a slightly changed model. The script expects 20 input parameters, which define the model. After creating the necessary input, the script also automatically calls ABAQUS to start the modelling. All input and output data is stored in a separate directory for further processing, and also to prevent accidental deleting or overwriting of the results. This allows the script to be started again and again with different parameters, generating many different models without supervision.The script had to be modified in order to include the “plume” that was later introduced into the model. This script can probably be strongly shortened and optimised, however due to time running out, this rough working version was used. 8. makeVenus.m

% 'makeVenus' assembles a global map of Venus from the 8x4 VICAR topography files clear close all clc % Four Bands made up from 8 files each Venus1=[]; Venus2=[]; Venus3=[]; Venus4=[]; figure hold on for i=1:8 data=VICAR(['F0' num2str(i) '.IMG']); % VICAR(file) reads a VICAR image into the matrix data Venus1(1:1024,((i-1)*1024+1:i*1024))=data; end; data=VICAR('F09.IMG'); Venus2=[Venus2 data]; for i=10:16 data=VICAR(['F' num2str(i) '.IMG']); Venus2=[Venus2 data]; end for i=17:24 data=VICAR(['F' num2str(i) '.IMG']); Venus3=[Venus3 data];

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end for i=25:32 data=VICAR(['F' num2str(i) '.IMG']); Venus4=[Venus4 data]; end Venus=[Venus1(2:1024,:);Venus2(2:1024,:);Venus3(2:1024,:);Venus4(2:1024,:)]; % display the map Lat=linspace(-66.5133,66.5133,4092); Lon=linspace(-120,240,8192); imagesc(Lon,Lat,Venus) set(gca,'YDir','normal') axis equal tight caxis([0 14000]) colorbar save Venusmap

makeVenus.m is the Venusian counterpart to makeearth.m. The Magellan topography data is portioned into 32 seperated segements, arranged in four rows of eight squares running parallel to the planets equator. These data parcels are provided in a NASA developed format called VICAR. The VICAR format is explained on the Magellan data homepage. From this information, a script was constructed, that reads the data into a matrix for each parcel, and connects these parcels to a complete Venusian topography map of the planet. This map is then saved and displayed. 9. mexearth.m

clear all close all clc %load ../venustopo/Artkm load Aleutkm %Artkm=Artkm/1000; data=Imageflat(Aleutkm,10); data=data/10; number=20;

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hatsize=160; [datax,datay]=size(data); hats=zeros(hatsize,hatsize,number); Data=zeros(datax+hatsize-1,datay+hatsize-1,number); Dominant=zeros(size(Data(:,:,1))); lengths=linspace(10,0.75*hatsize,number); disp('prepared') for i=1:number hats(:,:,i)=mex2D(lengths(i),hatsize); hats(:,:,i)=hats(:,:,i)./max(max(hats(:,:,i))); disp('hats done') temp=hats(:,:,i); temp2=abs(conv2(data,temp)); disp('convolution done') Data(:,:,i)=temp2/(max(max(temp2))); i end; subs=ceil(sqrt(number)); figure for i=1:number subplot(subs,subs,i) imagesc(Data(:,:,i)) colorbar end [pseud,Dominant]=max(Data,[],3); [x1,y1]=size(Dominant); [x2,y2]=size(data); x=round((x1-x2)/2); y=round((y1-y2)/2); lengths=lengths*10; Dominant=(Dominant(x:x1-x,y:y1-y)); Dominant=lengths(Dominant); figure imagesc(Dominant) colorbar %save Earthdominants figure

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imagesc(data) %caxis([-5000 0]) colorbar

the mexearth.m script performs the Mexican hat wavelet analysis on a provided input matrix of earth topography. The first step is to use use Imageflat to decrease the amount of data that has to be processed. The the appropriate hats are calculate and the resulting matrices stored in a 3D tensor. Of this tensor, the maximum values for each point are extracted and displayed in map form. The script also saves the results and displays the hats, the original data, and a before/after comparison of the smoothing process. A variation for venus is the script mexvenus.m.

9. mexihat2D.m function [out1] = mexihat(varargin) if errargn(mfilename,nargin,[3 4],nargout,[0:2]), error('*'); end out2 = linspace(varargin{1:3}); [X,Y]=meshgrid(out2,out2); out1=X.*X+Y.*Y; out1 = (2/(sqrt(3)*pi^0.25)) * exp(-out1/2) .* (1-out1); This script is a variation of the Matlab routine “mexihat”, that creates the 3D Mexican Hat used for the convolutions in the Wavelength analysis. 10. mygrid_sand.m

% Function MYGRID_SAND Read bathymetry data from Sandwell Database % [image_data,vlat,vlon] = mygrid_sand(region) % % program to get bathymetry from topo_8.2.img % WARNING: change DatabasesDir to the correct one for your machine % Catherine de Groot-Hedlin % latitudes must be between -72.006 and 72.006; % input: % region =[south north west east]; % output: % image_data - matrix of sandwell bathymetry/topography % vlat - vector of latitudes associated with image_data % vlon - vector of longitudes % function [image_data,vlat,vlon] = mygrid_sand(region)

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DatabasesDir = '/home/data_2/ralph/earthtopo/images/'; % determine the requested region blat = region(1); tlat = region(2); wlon = region(3); elon = region(4); % Setup the parameters for reading Sandwell data db_res = 2/60; % 2 minute resolution db_loc = [-72.006 72.006 0.0 360-db_res]; db_size = [6336 10800]; nbytes_per_lat = db_size(2)*2; % 2-byte integers image_data = []; % Determine if the database needs to be read twice (overlapping prime meridian) if ((wlon<0)&(elon>=0)) % wlon = [wlon 0]; % elon = [360-db_res elon]; wlon = [360+wlon 0]; elon = [360-db_res elon]; end % Calculate number of "records" down to start (latitude) (0 to db_size(1)-1) % (mercator projection) rad=pi/180;arg1=log(tan(rad*(45+db_loc(1)/2))); arg2=log(tan(rad*(45+blat/2))); iblat = fix(db_size(1) +1 - (arg2-arg1)/(db_res*rad)) arg2=log(tan(rad*(45+tlat/2))); itlat = fix(db_size(1) +1 - (arg2-arg1)/(db_res*rad)) if (iblat < 0 ) | (itlat > db_size(1)-1) errordlg([' Requested latitude is out of file coverage ']); end % Go ahead and read the database for i = 1:length(wlon); % Open the data file fid = fopen([DatabasesDir '/earthtopo_8.2.img'], 'r','b'); if (fid < 0) errordlg(['Could not open database: ' DatabasesDir '/earthtopo_8.2.img'],'Error'); end % Make sure the longitude data goes from 0 to 360 if wlon(i) < 0

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wlon(i) = 360 + wlon(i); end if elon(i) < 0 elon(i) = 360 + elon(i); end % Calculate the longitude indices into the matrix (0 to db_size(1)-1) iwlon(i) = fix((wlon(i)-db_loc(3))/db_res) ielon(i) = fix((elon(i)-db_loc(3))/db_res) if (iwlon(i) < 0 ) | (ielon(i) > db_size(2)-1) errordlg([' Requested longitude is out of file coverage ']); end % allocate memory for the data data = zeros(iblat-itlat+1,ielon(i)-iwlon(i)+1); % Skip into the appropriate spot in the file, and read in the data disp('Reading in bathymetry data'); for ilat = itlat:iblat offset = ilat*nbytes_per_lat + iwlon(i)*2; status = fseek(fid, offset, 'bof'); data(iblat-ilat+1,:)=fread(fid,[1,ielon(i)-iwlon(i)+1],'integer*2'); end % close the file fclose(fid); % put the two files together if necessary if (i>1) image_data = [image_data data]; else image_data = data; end end % Determine the coordinates of the image_data vlat=zeros(1,iblat-itlat+1); arg2 = log(tan(rad*(45+db_loc(1)/2.))); for ilat=itlat+1:iblat+1; arg1 = rad*db_res*(db_size(1)-ilat+0.5); term=exp(arg1+arg2); vlat(iblat-ilat+2)=2*atan(term)/rad -90; end vlon=db_res*((iwlon+1:ielon+1)-0.5); % to plot it up

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[xx,yy]=meshgrid(vlon,vlat); pcolor(xx,yy,image_data),shading flat,colormap(jet),colorbar('vert') xlabel('longitude'),ylabel('latitude'),title('Smith and Sandwell bathymetry')

This is the mygrid_sand.m script that is supplied together with the topography and bathymetry data used for this project. The script reads topographic data from the topo8_1.img data file according to the boundary coordinates that are supplied by the user. It also transforms the data into the suitable Mercator projection and labels the axes with the correct coordinates. 11. runmodels.m

close all clear all clc number=3 a=1:number; blocksize=[15000 18000 20000] platethick= [1] depth= [420000 432000 420000] width=[2400000 2412000 2400000] surfacetemp=[1000]; platetemp=728; sublength=[100000]; g=8.87; platedens=[3330]; mantledens=[3250]; subdens=[8000]; platepois=[0.25]; mantlepois=[0.35] ;

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Young=[10^11]; relax=[7]; WLF1=[6]; WLF2=1000; conduct=3; specheat=1500; plumetemp=[3500]; plumewidth=[900000]; plumethick=[6]; Profmatrix=[]; save Profmatrix Profmatrix; for i=1:length(a) makeinplume(a(i), depth(i), width(i), blocksize(i), surfacetemp, platetemp, platethick, subdens, sublength, g, platedens, mantledens, platepois, mantlepois, Young, relax, WLF1, WLF2, conduct, specheat, plumetemp, plumewidth, plumethick) end

runmodels.m is the the third script, together with makeinp.m and filconvert.m, that allows the modelling to be run with several parameters without supervision. This script is designed to supply makeinp.m with the necessary parameters to generate and run a model. The version above is modified to create input for the makeinplume.m variant.

All scripts above, with the exception of “mygrid_sand.m”, are freeware and may be used and modified as you choose. The scripts are not guaranteed to work.

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Appendix C – a typical ABAQUS input file *HEADING ** ABAQUS job created on 29-May-02 at 12:16:40 ** *INCLUDE, INP=Nodes.abq ** *INCLUDE, INP=Blockelements.abq ** *INCLUDE, INP=Material.abq ** *INCLUDE, INP=Box.abq ** *INCLUDE, INP=Temp.abq ** *INCLUDE, INP=SURP.abq ** *INCLUDE, INP=GravSet.abq ** *STEP,INC=800,AMPLITUDE=STEP,NLGEOM *VISCO, CETOL=0.0005, STABILIZE 6.3E+11, 3.1E+15, 1.E+7 ** *INCLUDE, INP=Gravity.abq ** *INCLUDE, INP=Output.abq ** *END STEP The basic input file. The include sub-files are shown separately below. *NODE 1, 0, 0 2, 15000, 0 3, 30000, 0 4, 45000, 0 5, 60000, 0 . . . . 4662, 2295000, 420000 4663, 2310000, 420000 4664, 2325000, 420000 4665, 2340000, 420000 4666, 2355000, 420000 4667, 2370000, 420000 4668, 2385000, 420000 4669, 2400000, 420000 ** The “Nodes.abq” include file which describes the positions of all the nodes used for the model, only beginning and end are shown, as the rest should be obvious. *ELEMENT, ELSET=BLOCK, TYPE=CAX4 1, 1, 2, 163, 162

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2, 2, 3, 164, 163 3, 3, 4, 165, 164 4, 4, 5, 166, 165 5, 5, 6, 167, 166 6, 6, 7, 168, 167 . . . . 4473, 4500, 4501, 4662, 4661 4474, 4501, 4502, 4663, 4662 4475, 4502, 4503, 4664, 4663 4476, 4503, 4504, 4665, 4664 4477, 4504, 4505, 4666, 4665 4478, 4505, 4506, 4667, 4666 4479, 4506, 4507, 4668, 4667 4480, 4507, 4508, 4669, 4668 ** Blockelements.abq, the input file that describes the relation between nodes and elements in the model. *SOLID SECTION,MATERIAL=STUFF,ELSET=BLOCK *MATERIAL,NAME=STUFF *DENSITY 8000., 400 3380., 700 3330., 1000 3300., 1001 3200., 4500 *ELASTIC,TYPE=ISO,MODULI=INSTANTANEOUS 10.00E+10, 0.25, 700 10.00E+10, 0.26, 1000 10.00E+10, 0.34, 1001 10.00E+10, 0.35, 4500 *EXPANSION,ZERO=0.,TYPE=ISO 3.1E-5, *VISCOELASTIC,TIME=PRONY 1., 0., 2.10E+8 *TRS,DEFINITION=WLF 1000.0,6,1000 *CONDUCTIVITY 3.0 *SPECIFIC HEAT 1500.0 Material.abq, which describe the properties of the material used in the model. ** floor *BOUNDARY, OP=MOD 1, 2,, 0 2, 2,, 0 3, 2,, 0 4, 2,, 0 . . . . 158, 2,, 0 159, 2,, 0

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160, 2,, 0 161, 2,, 0 ** ** sides *BOUNDARY, OP=NEW 1, 1,, 0 161, 1,, 0 162, 1,, 0 322, 1,, 0 323, 1,, 0 483, 1,, 0 . . . . 4348, 1,, 0 4508, 1,, 0 4509, 1,, 0 4669, 1,, 0 ** Box.abq is the include file which determines the sides and floor of the model, as well as the boundary movement conditions. *NSET,NSET=SURP,GENERATE 4509, 4669, 1 ** SURP.abq simply generates the element set for the surface profile to be extracted. ** Gravity *ELSET, GENERATE, ELSET=GRAVITY 1, 4480, 1 ** Gravset.abq defines the element set that gravity is applied to (all elements). *DLOAD, OP=NEW GRAVITY, GRAV, 8.87, 0., -1., 0. ** Gravity.abq is the input file that determines the direction and strength of gravity in the model. *Output, Field, OP=NEW, frequency=2 *Node output COORD, U, RF, CF, NT, RFL *Element output S, CE, E, EE, ER, PEEQ, ENER, DENSITY *FILE FORMAT, ASCII *NODE FILE,FREQUENCY=100,NSET=SURP COORD ** Output.abq selectes the parameters and format of the desired ASCII output.

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References: Avduevsk.Vs, M. Y. Marov, et al. (1971). "Soft Landing of Venera-7 on Venus

Surface and Preliminary Results of Investigations of Venus Atmosphere." Journal of

the Atmospheric Sciences 28(2): 263-&.

Barsukov, V. L., A. T. Bazilevskii, et al. (1984). "1st Results of Geologomorphological

Analysis of the Venus Surface Radiolocation Images Obtained from the Venera-15

and Venera-16 Ais." Doklady Akademii Nauk Sssr 279(4): 946-&.

Brunet, D. and D. A. Yuen (2000). "Mantle plumes pinched in the transition zone."

Earth and Planetary Science Letters 178(1-2): 13-27.

Burba, G. A. (1990). "Names on the Maps of Venus - a Pre-Magellan Review." Earth

Moon and Planets 50-1: 541-558.

Carpente.Rl (1970). "A Radar Determination of Rotation of Venus." Astronomical

Journal 75(1): 61-&.

Curtis, H. H. (1994). "Magellan - Aerobraking at Venus." Aerospace America 32(1):

32-&.

Dyer, J. W., Nunamake.Rr, et al. (1974). "Pioneer Venus Mission Plan for

Atmospheric Probes and an Orbiter." Journal of Spacecraft and Rockets 11(10): 710-

715.

Dziewonski, A. M. and D. L. Anderson (1981). "Preliminary Reference Earth Model."

Physics of the Earth and Planetary Interiors 25(4): 297-356.

Goldstei.Rm (1965). "Preliminary Venus Radar Results." Journal of Research of the

National Bureau of Standards Section D-Radio Science D 69(12): 1623-&.

98

Hoernle, K., Zhang, Y-S. and Graham, D. (1995) “Seismic and geochemical evidence

for large-scale mantle upwelling beneath the eastern Atlantic and western and central

Europe” Nature 374, 34-39

Ingersoll, A. P. and A. R. Dobrovolskis (1978). "Venus Rotation and Atmospheric

Tides." Nature 275(5675): 37-38.

Ivanov, M. A. and J. W. Head (1996). "Tessera terrain on Venus: A survey of the

global distribution, characteristics, and relation to surrounding units from Magellan

data." Journal of Geophysical Research-Planets 101(E6): 14861-14908.

Kiefer, W. S. and B. H. Hager (1991). "A Mantle Plume Model for the Equatorial

Highlands of Venus." Journal of Geophysical Research-Planets 96(E4): 20947-

20966.

Kotelnikov, V. A., E. L. Akim, et al. (1984). "The Maxwell Montes Region, Surveyed

by the Venera 15, Venera 16 Orbiters." Soviet Astronomy Letters 10(6): 369-376.

Lyons, D. T., R. S. Saunders, et al. (1995). "The Magellan Venus Mapping Mission -

Aerobraking Operations." Acta Astronautica 35(9-11): 669-676.

Malamud, B. D. and D. L. Turcotte (2001). "Wavelet analyses of Mars polar

topography." Journal of Geophysical Research-Planets 106(E8): 17497-17504.

McNamee, J. B., N. J. Borderies, et al. (1993). "Venus - Global Gravity and

Topography." Journal of Geophysical Research-Planets 98(E5): 9113-9128.

Pettengill, G. H., P. G. Ford, et al. (1979). "Venus - Preliminary Topographic and

Surface Imaging Results from the Pioneer Orbiter." Science 205(4401): 90-93.

Prinn, R. G. (1973). "Venus - Composition and Structure of Visible Clouds." Science

182(4117): 1132-1135.

99

Schubert, G., W. B. Moore, et al. (1994). "Gravity over Coronae and Chasmata on

Venus." Icarus 112(1): 130-146.

Smith, W. H. F. and D. T. Sandwell (1997). "Global sea floor topography from

satellite altimetry and ship depth soundings." Science 277(5334): 1956-1962.

Squyres, S. W., D. M. Janes, et al. (1992). "The Morphology and Evolution of

Coronae on Venus." Journal of Geophysical Research-Planets 97(E8): 13611-13634.

Books used:

Brémaud, P., Mathematical Principles of Signal Processing, 2002, Springer New York,

pp.269

Cattermole, P., Venus – The Geological Story, 1994, UCL Press, pp.250

Christiansen, E. H., Exploring the Planets, Second Edition, 1995, Prentice Hall, pp.500

Ferry, J.D., Viscoelastic Properties of Polymers, 1980, pp. 641

Ford, J.P., Plaut, J.J., Weitz, C.M., Farr, T.G., Senske, D.A., Stofan, E.R., Michaels, G.,

Parker, T.J., Guide to Magellan Image Interpretation, 1993, JPL Publication

93-24, pp.147

Lowrie, W., Fundamentals of Geophysics, 1997, Cambridge University Press, pp.354

Marev, M. Y. and Grinspoon, D. H., The Planet Venus, 1998, Yale University Press, pp.442

Morrison, D., Planetenwelten – Eine Entdeckungsreise durch das Sonnensystem, 1992,

Spektrum Akademischer Verlag, pp.238

Murray, B., Malin, M.C., Greenly, R., Earthlike Planets – Surfaces of Mercury, Venus,

Earth, Moon, Mars, 1981, W.H. Freeman & Company, pp.387

100

Press, F. and Siever, R., Understanding Earth, 1998, W.H. Freeman Press, pp.121

Stöcker, H., Taschenbuch der Physik, 1998, Verlag Harri Deutsch, pp.1091

Uchupi, E. and Emery, K., Morphology of the Rocky Members of the Solar System, 1993,

Springer New York, pp.395

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